Self-similar and fractal design for stretchable electronics

ABSTRACT

The present invention provides electronic circuits, devices and device components including one or more stretchable components, such as stretchable electrical interconnects, electrodes and/or semiconductor components. Stretchability of some of the present systems is achieved via a materials level integration of stretchable metallic or semiconducting structures with soft, elastomeric materials in a configuration allowing for elastic deformations to occur in a repeatable and well-defined way. The stretchable device geometries and hard-soft materials integration approaches of the invention provide a combination of advance electronic function and compliant mechanics supporting a broad range of device applications including sensing, actuation, power storage and communications.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. patent application No. U.S.Ser. No. 14/766,333, which has a § 371(c) and national entry date ofAug. 6, 2015, which is a National Stage Application under 35 U.S.C. §371 of International Application No. PCT/US2014/014932, filed Feb. 5,2014, which claims the benefit of and is a continuation-in-part of U.S.patent application Ser. No. 13/835,284, filed on Mar. 15, 2013, U.S.Provisional Patent Application 61/761,412, filed on Feb. 6, 2013, andU.S. Provisional Patent Application 61/930,732, filed on Jan. 23, 2014,each of which is hereby incorporated by reference in its entirety.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

This invention was made with government support under contract numberDE-FG02-07ER46471 awarded by the US Department of Energy. The governmenthas certain rights in the invention.

BACKGROUND

Since the first demonstration of a printed, all polymer transistor in1994, a great deal of interest has been directed at a potential newclass of electronic systems comprising flexible integrated electronicdevices on plastic substrates. [Garnier, F., Hajlaoui, R., Yassar, A.and Srivastava, P., Science, Vol. 265, pgs 1684-1686] Recently,substantial research has been directed toward developing new solutionprocessable materials for conductors, dielectrics and semiconductors forflexible plastic electronic devices. Progress in the field of flexibleelectronics, however, is not only driven by the development of newsolution processable materials but also by new device componentgeometries, efficient device and device component processing methods andhigh resolution patterning techniques applicable to flexible electronicsystems. It is expected that such materials, device configurations andfabrication methods will play an essential role in the rapidly emergingnew class of flexible integrated electronic devices, systems andcircuits.

Interest in the field of flexible electronics arises out of severalimportant advantages provided by this technology. For example, theinherent flexibility of substrate materials allows them to be integratedinto many shapes providing for a large number of useful deviceconfigurations not possible with brittle conventional silicon basedelectronic devices. In addition, the combination of solution processablecomponent materials and flexible substrates enables fabrication bycontinuous, high speed, printing techniques capable of generatingelectronic devices over large substrate areas at low cost.

The design and fabrication of flexible electronic devices exhibitinggood electronic performance, however, present a number of significantchallenges. First, the well-developed methods of making conventionalsilicon based electronic devices are incompatible with most flexiblematerials. For example, traditional high quality inorganic semiconductorcomponents, such as single crystalline silicon or germaniumsemiconductors, are typically processed by growing thin films attemperatures (>1000 degrees Celsius) that significantly exceed themelting or decomposition temperatures of most plastic substrates. Inaddition, most inorganic semiconductors are not intrinsically soluble inconvenient solvents that would allow for solution based processing anddelivery. Further, although many amorphous silicon, organic or hybridorganic-inorganic semiconductors are compatible with incorporation intoflexible substrates and can be processed at relatively low temperatures,these materials do not have electronic properties capable of providingintegrated electronic devices capable of good electronic performance.For example, thin film transistors having semiconductor elements made ofthese materials exhibit field effect mobilities approximately threeorders of magnitude less than complementary single crystalline siliconbased devices. As a result of these limitations, flexible electronicdevices are presently limited to specific applications not requiringhigh performance, such as use in switching elements for active matrixflat panel displays with non-emissive pixels and in light emittingdiodes.

Flexible electronic circuitry is an active area of research in a numberof fields including flexible displays, electro-active surfaces ofarbitrary shapes such as electronic textiles and electronic skin. Thesecircuits often are unable to sufficiently conform to their surroundingsbecause of an inability of the conducting components to stretch inresponse to conformation changes. Accordingly, those flexible circuitsare prone to damage and electronic degradation and can be unreliableunder rigorous and/or repeated conformation change. Flexible circuitsrequire stretchable and bendable interconnects that remain intact whilecycling through stretching and relaxation.

Conductors that are capable of both bending and elasticity are generallymade by embedding metal particles in an elastomer such as silicone.Those conductive rubbers are both mechanically elastic and electricallyconductive. The drawbacks of a conductive rubber include high electricalresistivity and significant resistance changes under stretching, therebyresulting in overall poor interconnect performance and reliability.

Gray et al. discuss constructing elastomeric electronics usingmicrofabricated tortuous wires encased in a silicone elastomer capableof linear strains up to 54% while maintaining conductivity. In thatstudy, the wires are formed as a helical spring-shape. In contrast tostraight-line wires that fractured at low strains (e.g., 2.4%), tortuouswires remained conductive at significantly higher strains (e.g., 27.2%).Such a wire geometry relies on the ability of wires to elongate bybending rather than stretching. That system suffers limitations in theability to controllably and precisely pattern in different shapes and inadditional planes, thereby limiting the ability to tailor systems todifferent strain and bending regimes.

Studies suggest that elastically stretchable metal interconnectsexperience an increase in resistance with mechanical strain. (Mandlik etal. 2006). Mandlik et al. attempt to minimize this resistance change bydepositing metal film on pyramidal nanopatterned surfaces. That study,however, relies on the relief feature to generate microcracks thatimpart stretchability to thin metal lines. The microcracks facilitatemetal elastic deformation by out of plane twisting and deformation.Those metal cracks, however, are not compatible with thick metal films,and instead are compatible with a rather narrow range of thin metalfilms (e.g., on the order of less than 30 nm) that are deposited on topof patterned elastomers.

One manner of imparting stretchability to metal interconnects is byprestraining (e.g., 15%-25%) the substrate during conductor (e.g.,metal) application, followed by spontaneous relief of the prestain,thereby inducing a waviness to the metal conductor interconnects. (see,e.g., Lacour et al. (2003); (2005); (2004), Jones et al. (2004); Huck etal. (2000); Bowden et al. (1998)). Lacour et al. (2003) report byinitially compressing gold stripes to generate spontaneously wrinkledgold stripes, electrical continuity is maintained under strains of up to22% (compared to fracture strains of gold films on elastic substrates ofa few percent). That study, however, used comparatively thin layers ofmetal films (e.g., about 105 nm) and is relatively limited in that thesystem could potentially make electrical conductors that could bestretched by about 10%.

U.S. Pat. Nos. 7,557,367, 7,521,292, and 8,217,381 and US PatentPublication Nos. 2010/0002402, 2012/0157804, and 2011/0230747 describeflexible and/or stretchable electronic systems accessed bymicrofabrication pathways including printing-based techniques. Thestretchable systems of these references include devices havingdistributed electronic device components interconnected via deformableelectronic interconnects, optionally capable of elastic responses tolarge strain deformation. The systems of these references includeelectronic devices for applications including tissue mounted biomedicaldevices, solar energy and large area macroelectronic systems.

From the forgoing, it is apparent there is a need for electronic devicessuch as interconnects and other electronic components having improvedstretchability, electrical properties and related processes for rapidand reliable manufacture of stretchable interconnects in a variety ofdifferent configurations. Progress in the field of flexible electronicsis expected to play a critical role in a number of important emergingand established technologies. The success of these applications offlexible electronics technology depends strongly, however, on thecontinued development of new materials, device configurations andcommercially feasible fabrication pathways for making integratedelectronic circuits and devices exhibiting good electronic, mechanicaland optical properties in flexed, deformed and bent conformations.Particularly, high performance, mechanically extensible materials anddevice configurations are needed exhibiting useful electronic andmechanical properties in folded, stretched and/or contractedconformations.

SUMMARY

The present invention provides electronic circuits, devices and devicecomponents including one or more stretchable components, such asstretchable electrical interconnects, electrodes and/or semiconductorcomponents. Stretchability of some of the present systems is achievedvia a materials level integration of stretchable metallic orsemiconducting structures with soft, elastomeric materials in aconfiguration allowing for elastic deformations to occur in a repeatableand well-defined way. The stretchable device geometries and hard-softmaterials integration approaches of the invention provide a combinationof advance electronic function and compliant mechanics supporting abroad range of device applications including sensing, actuation, powerstorage and communications. Specific classes of devices benefiting fromthe enhanced mechanical properties of the present systems includestretchable batteries, radio frequency antennas, tissue mountedelectronics and sensors compatible with magnetic resonance imaging.

In some aspects, the invention provides a new class of stretchablesystems having specific geometries which achieve enhancements overconventional flexible and stretchable devices for accommodating elasticstrain. In embodiments, for example, stretchable metallic orsemiconducting structures of the invention are characterized by a twodimensional geometry characterized by a plurality of spatialfrequencies, for example, via a spring-in-a-spring overall geometrycapable of supporting a wide range of deformation modes without failureor significant degradation of performance. Specific device architecturesof the present invention include stretchable metallic or semiconductingstructures having self-similar geometries and/or fractal-like geometriescapable of supporting a range of biaxial and/or radial deformationmodes, thereby, providing a versatile device platform for a range ofstretchable electronic and/or optical systems.

In an aspect, the present invention provides electronic circuitsexhibiting stretchability. In an embodiment, for example, an electroniccircuit comprises: an elastic substrate; and a stretchable metallic orsemiconducting device component supported by the elastic substrate; thestretchable metallic or semiconducting device component comprising aplurality of electrically conductive elements each having a primary unitcell shape, the electrically conductive elements connected in a sequencehaving a secondary shape providing an overall two-dimensional spatialgeometry characterized by a plurality of spatial frequencies; whereinthe two-dimensional spatial geometry of the metallic or semiconductingdevice component allows for accommodation of elastic strain along one ormore in-plane or out of plane dimensions (or directions), therebyproviding stretchability of the electronic circuit. In some embodiments,for example, at least a portion of the stretchable metallic orsemiconducting device component is in physical contact with the elasticsubstrate or an intermediate structure provided between the substrateand the stretchable metallic or semiconducting device component. In anembodiment, the electronic circuit of the invention comprises astretchable electronic device, semiconductor device, device array orcomponent thereof, for example, comprising a plurality of metallic orsemiconducting device component supported by the elastomeric substrate.

In embodiments, the two-dimensional spatial geometry of the metallic orsemiconducting device component(s) allows for significant deformationwithout substantial degradation of electronic performance or failure,for example, via compression, expansion, twisting and/or bendingdeformations. In an embodiment, for example, the two-dimensional spatialgeometry allows the metallic or semiconducting device component toundergo elastic deformation. In an embodiment, for example, thetwo-dimensional spatial geometry allows the metallic or semiconductingdevice component to undergo biaxial deformation, radial deformation orboth. In an embodiment, for example, the two-dimensional spatialgeometry allows the metallic or semiconducting device component toundergo in-plane deformation, out-of-plane deformation or both.

Systems of the invention include metallic or semiconducting devicecomponent(s) having a broad range of two-dimensional spatial geometriesproviding enhanced stretchability. Aspects of the two-dimensionalspatial geometry for some embodiments exhibit a self-similarcharacteristic, such as a spatial geometry exhibiting an iterativepattern or pattern of patterns, for example, provided in aspring-within-a-spring type configuration. Metallic or semiconductingdevice component(s) of the present systems having self-similar and/orfractal-based geometries provide a beneficial combination of high fillfactors and useful mechanical stretchability, for example, so as toprovide high area conformal coverage of surfaces having a complextopography, such as curved surfaces (e.g., radius of curvature greaterthan or equal to 0.01 mm, optionally 0.1 mm) and surfaces characterizedby one or more relief or recessed features. Some stretchable electroniccircuits of the invention are capable of high area conformal coverage ofthe surface of a biological tissue, such as, for example, a tissuesurface characterized by a complex morphology.

In exemplary embodiments, the two-dimensional spatial geometry of themetallic or semiconducting device component(s) is characterized by afirst spatial frequency having a first length scale corresponding to theprimary unit cell shape and a second spatial frequency having a secondlength scale corresponding to the secondary shape, for example, that ismade up of a sequence of elements having the primary unit cell shape. Insome embodiments, the first length scale is substantially different thanthe second length scale. For example, in the embodiment in FIG. 1C, thefirst length scale (L₁) is approximately 27% that of the second lengthscale (L₂); further, the first length scale (L₁) is approximately 8%that of the third length scale. More generally in this example, theratio L_(N)/L_(N+1) falls between 0.27 and 0.34, where N is the N^(th)length scale. This ratio L_(N)/L_(N−1) can be generally tailored; FIG.1B elements 111 and 112 show two examples of a self-similar serpentinehorseshoe pattern in which L_(N)/L_(N+1) is approximately 0.26 and 0.18,respectively.

For example, in one embodiment, first length scale of the first spatialfrequency is at least 2 times smaller than the second length scale ofthe second spatial frequency, and optionally for some embodiments thefirst spatial frequency is at least 5 times smaller than the secondlength scale of the second spatial frequency, the first spatialfrequency is at least 10 times smaller than the second length scale ofthe second spatial frequency. For example, in an embodiment, the firstlength scale of the first spatial frequency is 2 to 10 times smallerthan the second length scale of the second spatial frequency andoptionally for some embodiments, the first length scale of the firstspatial frequency is 3 to 5 times smaller than the second length scaleof the second spatial frequency. In a specific embodiment, the firstlength scale of the first spatial frequency is selected from the rangeof 100 nm to 1 mm and the second length scale of the second spatialfrequency is selected over the range of 1 micron to 10 mm. In exemplaryembodiments, the plurality of spatial frequencies are furthercharacterized by a plurality of length scales (e.g., first, second,third, etc.) characterized by a power series.

In some embodiments, the two-dimensional spatial geometry of thestretchable metallic or semiconducting device component is characterizedby more than two spatial frequencies, for example, having third, fourth,fifth, etc. spatial frequencies. For example, in one embodiment, thesequence of electrically conductive elements is further characterized bya tertiary shape comprising a repeating series of the electricallyconductive elements comprising the secondary shape. In embodiments, thetwo-dimensional spatial geometry is characterized by 2 to 5 spatialfrequencies. Use of a two-dimensional spatial geometry characterized bymore than two spatial frequencies is beneficial in some embodiments forproviding enhance areal coverage, stretchability and/or electricalresistance, for example, for antenna and radio frequency deviceapplications.

In embodiments, the two-dimensional spatial geometry is an iterativetwo-dimensional geometry, such as a spatial geometry comprising arepeating pattern of elements having the same or similar shape (e.g.,characterized by the unit cell shape). In an embodiment, for example,sequence of electrically conductive elements has a serial configurationor a branched configuration. Two-dimensional spatial geometries usefulin the present systems may have a deterministic two-dimensional shape ora random two-dimensional shape. Optionally, for some embodiments, thetwo-dimensional spatial geometry has a spring-within-a-spring geometry.For example, in embodiments, the spring-within-in-spring geometrycomprises a series of primary spring structures each independentlyhaving the primary unit cell shape connected in a serial or a branchedconfiguration to form one or more secondary spring structures eachindependently having the secondary shape. Optionally, the primary springstructures, the secondary spring structures or both comprise acompression spring structure or coiled spring structure. In someembodiments, the spring-in-a-spring geometry is characterized bytwo-dimensional geometry comprising at least two spring configurations,wherein upon deformation a first spring configuration undergoesstretching to some degree prior to stretching of a second springconfiguration.

In some embodiments, the two-dimensional spatial geometry is aself-similar two-dimensional geometry. For example, in embodiments, theself-similar two-dimensional geometry is characterized by the secondaryshape being similar to the primary unit cell shape but having adifferent length scale. In some embodiments, the self-similartwo-dimensional geometry is characterized by a length scale of thesecondary shape at least 2 times larger than a length scale of theprimary unit cell shape, optionally at least 3 times larger than alength scale of the primary unit cell shape. For example, inembodiments, a length scale of the secondary shape is larger than alength scale of the primary unit cell shape by a factor selected overthe range of 2 to 20, and optionally selected over the range of 3 to 10.In an embodiment, a self-similar geometry is characterized by a primaryunit cell having a unit cell shape and one or more higher order patternsmade up of the primary unit cell and having a similar overall spatialconfiguration as the primary unit cell shape but with a different lengthscale. In some embodiments, for example, the length scale of the unitcell shape and the length scales of high order patterns are multiples ofeach other (e.g., related by a constant), thereby giving rise to anoverall two-dimensional geometry characterized by a plurality of spatialfrequencies.

In some embodiments, the self-similar two-dimensional geometry ischaracterized by a first spatial frequency having a first length scalecorresponding to the primary unit cell shape, a second spatial frequencyhaving a second length scale corresponding to the secondary shape and athird spatial frequency having a third length scale corresponding to atertiary shape. Optionally, the tertiary shape is similar to the primaryunit cell shape and the secondary shape but having a different lengthscale. For example, in an embodiment, the first length scale of thefirst spatial frequency is 3 to 50 times smaller than the third lengthscale of the third spatial frequency and optionally for someembodiments, and the second length scale of the second spatial frequencyis 2 to 10 times smaller than the third length scale of the thirdspatial frequency. In a specific embodiment, the first length scale ofthe first spatial frequency is selected from the range of 100 nm to 1 mmand the second length scale of the second spatial frequency is selectedover the range of 1 micron to 10 mm, and the third length scale of thethird spatial frequency is selected over the range of 10 micron to 100mm. In exemplary embodiments, the plurality of spatial frequencies arefurther characterized by a plurality of length scales (e.g., first,second, third, etc.) characterized by a power series. The systems of theinvention also include self-similar two-dimensional geometriescharacterized by even high orders of self-similar shape (e.g., 4, 5, 6,etc. orders).

In some embodiments, for example, the two-dimensional spatial geometryis a fractal-based two-dimensional geometry, for example, characterizedby a fractal dimension greater than or equal to 1, optionally for someapplications greater than or equal to 1.3, optionally for someapplications greater than or equal to 1.5. In some embodiments, thetwo-dimensional spatial geometry is a fractal-based two-dimensionalgeometry characterized by a fractal dimension selected from the range of1 to 2. Use of a self-similar and/or a fractal-based two-dimensionalgeometries for stretchable metallic or semiconducting device componentsis beneficial for providing systems capable of undergoing larger elasticstrains relative to convention serpentine systems.

In some embodiments, the two-dimensional spatial geometry of thestretchable metallic or semiconducting device component does not have aserpentine or mesh geometry characterized by only a single spatialfrequency. In some embodiments, the two-dimensional spatial geometry ofthe stretchable metallic or semiconducting device component does nothave serpentine or mesh geometry characterized by a rectangular orsquare secondary shape.

In some embodiments, the two-dimensional spatial geometry of thestretchable metallic or semiconducting device component provides a highfill factor between first and second device components or over an activearea of the electronic circuit, for example, a fill factor greater thanor equal to 25%, optionally for some applications greater than or equalto 50%, and optionally for some applications greater than or equal to70% For example, in some embodiments, the two-dimensional spatialgeometry of the stretchable metallic or semiconducting device componentprovides a fill factor between first and second device components orprovided over an active area of the electronic circuit selected from therange of 25% to 90%, optionally for some applications 50% to 90%

Embodiments of the invention utilize one or multiple structures providedin electrical communication with one another. For example, in oneembodiment, multiple components are provided in a serial configurationor a branched configuration to provide electrical communication betweentwo or more components of an electronic circuit, device or devicecomponent. For some embodiments, electrical communication is provided byone or more conductive or semiconductive elements, each independentlyhaving a two dimensional geometry characterized by a plurality ofspatial frequencies. In some embodiments, the flexibility, fabricationand reliability of electronic circuits of the invention are enhanced byuse of a single continuous structure, such as a unitary structure, or aplurality of discrete continuous structures, such as individual unitarystructures. Use of multiple structures optionally provides flexibilityand fabrication enhancements for other embodiments. For example, in someembodiments, a single unit-cell type geometry is fabricated in a largequantity and later assembled in a series or sequence to provide anelectronic circuit secondary shape. In other embodiments, multipledistinct unit-cell geometries are assembled to create a largerelectrical circuit characterized by a secondary shape.

In embodiments, for example, the electrically conductive elements of themetallic or semiconducting device component comprise a continuousstructure. For example, in an embodiment, the electrically conductiveelements of the metallic or semiconducting device component comprise asingle unitary structure, such as a monolithic structure comprising ametallic or semiconducting material. In an embodiment, for example, theelectrically conductive elements of the metallic or semiconductingdevice component comprise one or more thin film structures, for example,thin film structures generated by deposition (physical vapor deposition,chemical vapor deposition, atomic layer deposition, etc.), epitaxialgrowth and/or printing-based assembly techniques. In an embodiment, forexample, the electrically conductive elements of the metallic orsemiconducting device component comprise a continuous and monolithicthin film structure.

In some embodiments, the electrically conductive elements are one ormore of: free standing structures at least partially supported by theelastic substrate; tethered structures at least partially connected tothe elastic substrate; bound structures at least partially bound to theelastic substrate; embedded structures at least partially embedded inthe elastic substrate or in an embedding layer supported by thesubstrate; or structures within a containment structure and in physicalcontact with a containment fluid or a containment solid. In someembodiment, only a portion (e.g., less than or equal to 10%) of theelectrically conductive element is bonded to the elastic substrate or anintermediate structure provided between the stretchable metallic orsemiconducting device component and the elastic substrate. In someembodiments, at least a portion of the electrically conductive elementsof the metallic or semiconducting device component are provide within acontainment structure, for example, a containment structure having acontainment liquid or containment solid material having a low Young'smodulus (e.g., a Young's modulus less than or equal to 1 MPa, or lessthan or equal to 0.5 MPa, or less than or equal to 200 KPa). Containmentstructures of the invention include those formed by a substrate and asuperstrate arranged so as to accommodating a containment fluid or acontainment solid. In some embodiments, the devices and devicecomponents of the invention further comprise a low modulus intermediatelayer (e.g., a Young's modulus less than or equal to 1 MPa, or less thanor equal to 0.5 MPa, or less than or equal to 200 KPa) provided betweenthe substrate and the metallic or semiconducting device component, andoptionally at least partially in physical contact with, or bonded to,the metallic or semiconducting device component. In an embodiment ofthis aspect, the low modulus intermediate layer has a thickness lessthan or equal to 1000 μm, or less than or equal to 500 μm, or less thanor equal to 250 μm, or less than or equal to 100 μm.

Electrically conductive elements comprising a wide range of materialsand having a wide range of physical properties are useful in the presentinvention. In certain embodiments, each of the electrically conductiveelements independently has a thickness selected from the range of 10 nmto 1 mm, optionally for some applications selected from the range of 10nm to 100 μm, and optionally for some applications selected from therange of 10 nm to 10 μm In some embodiments, each of the electricallyconductive elements independently has a thickness less than or equal to1 micron, optionally for some applications less than or equal to 500 nm.Optionally, each unit cell shape of the electrically conductive elementsis independently characterized by lateral dimensions (e.g., length,width, radius, etc.) selected from the range of 100 nm to 10 mm.

In some embodiments, the stretchable metallic or semiconducting devicecomponents of systems of the invention are characterized by pathlengthsbetween first and second ends independently selected from the range of0.1 μm to 100 cm, optionally for some embodiments pathlengths selectedfrom the range of 1 μm to 1 mm. As used herein, pathlength refers to theminimum distance between first and second ends along the path of thestretchable metallic or semiconducting device component tracing the twodimensional geometry, for example, along the circuitous path of astretchable interconnect from first and second interconnected devicecomponents or along the along the circuitous path from first and secondends of a stretchable electrode. In an embodiment, for example, the oneor more electrical interconnects provide for electrical connectionbetween two connection points separated by a straight line distanceselected from the range of 0.1 μm to 100 cm. In an embodiment, thestretchable metallic or semiconducting device component provides forelectrical connection between first and second connection points andindependently has pathlengths at least 2 times larger than the shorteststraight line distance between the first and second connection points,and optionally at least 4 times larger than the shortest straight linedistance between the first and second connection points. In anembodiment, the stretchable metallic or semiconducting device componentprovides for electrical connection between first and second connectionpoints and independently has pathlengths 2 to 6 times larger than theshortest straight line distance between the first and second connectionpoints.

In exemplary embodiments, the electrically conductive elements comprisea wire, a ribbon or nanomembrane. In some embodiments, the electricallyconductive elements independently comprise a metal, an alloy, a singlecrystalline inorganic semiconductor or an amorphous inorganicsemiconductor. Use of single crystalline materials, doped materialsand/or high purity materials (e.g., purity great than or equal to 99.9%,optionally 99.99% and optionally 99.999% pure) for electricallyconductive elements is useful for certain device embodiments. In aspecific embodiment, the primary unit cell shape of the electricallyconductive elements comprises a spring, a fold, a loop, a mesh or anycombinations of these.

In some embodiments, for example, the primary unit cell shape comprisesa plurality of the spatially offset features, such as opposing segments,adjacent segments or a combination of opposition and adjacent segments.For example, in one embodiment, the spatially offset features of theprimary unit cell shape comprise a plurality of convex segments, concavesegments, circular segments, ellipsoidal segments, triangular segments,rectangular segments, square segments, or any combination of these.Optionally, the convex segments, concave segments, circular segments,ellipsoidal segments, triangular segments, rectangular segments orsquare segments of the unit cell shape are separated by one or morestraight line segments. Spatially offset features of the primary unitcell may be provided in an overall in-plane geometry wherein thefeatures provide a spatially variance within a plane parallel to asupporting surface of the substrate. Alternatively, spatially offsetfeatures of the primary unit cell may be provided in an overallout-plane geometry wherein the features provide spatial variance withina plane orthogonal to a supporting surface of the substrate. Inexemplary embodiments, the primary unit cell shape of the electricallyconductive elements is selected from the group consisting of one or moreof a von Koch curve, a Peano curve, a Hilbert curve, a Moore curve, aVicsek fractal, a Greek cross, or any combination of these.

In specific embodiments, for example, the stretchable metallic orsemiconducting device component has a curved configuration characterizedby a plurality of curved features, optionally provided within a planeparallel to a supporting surface, for example, provided in a periodic,serpentine, branched of mesh geometry. In specific embodiments, forexample, the stretchable metallic or semiconducting device component hasa bent, buckled or wrinkled configuration characterized by a pluralityof curved features, optionally provided within a plane orthogonal to asupporting surface, for example, provided in a periodic geometry.

In specific embodiments, for example, the stretchable metallic orsemiconducting device component comprises an electrode or an electrodearray. In one embodiment, the electrode or the electrode array is acomponent of sensor, actuator, or a radio frequency device. In anembodiment, for example, the electrode or the electrode array provides afill factor over an active area of the electronic circuit selected fromthe range of 25% to 90%, optionally for some applications 50% to 90% andoptionally for some applications 70% to 90%.

Aspects of some electronic circuit embodiments of the invention areuseful for providing electrical communication between other electroniccircuits and/or circuit components. For example, in embodiments, thestretchable metallic or semiconducting device component comprises one ormore electrical interconnects. Use of the electronic circuits ofembodiments of the invention as electrical interconnects gives theability to provide stretchability to conventional or rigid circuits bycontributing the beneficial stretchability aspects of the electroniccircuits of embodiments of the invention as interconnection pointsbetween conventional or rigid circuit components. For example, in someembodiments, an electronic circuit further comprises one or more rigidisland structures, wherein at least a portion of the one or moreelectrical interconnects is in electrical contact with the one or morerigid island structures. In one embodiment, for example, the rigidisland structures comprise inorganic semiconductor devices or devicecomponents. In embodiments, an electronic circuit further comprises aplurality of the stretchable metallic or semiconducting device componentin electrical contact with each of the rigid island structures. Forexample, in embodiments, the inorganic semiconductor devices or devicecomponents comprise a transistor, a diode, an amplifier, a multiplexer,a light emitting diode, a laser, a photodiode, an integrated circuit, asensor, a temperature sensor, a thermistor, a heater, a resistiveheater, an actuator or any combination of these.

Various elastic substrates are useful with embodiments of the inventionto provide support and stretchability for the electronic circuitcomponents of the system. In certain embodiments, the elastic substratehas an average thickness selected from the range of 0.25 μm to 10,000μm, optionally for some applications an average thickness selected fromthe range of 10 μm to 1,000 μm For example, in one embodiment, theelastic substrate has an average thickness less than or equal to 1000μm. Optionally, the elastic substrate is a prestrained elasticsubstrate. Useful elastic substrates include those having a Young'smodulus selected from the range of 0.5 KPa to 100 GPa, optionally forsome applications selected from the range of 10 KPa to 100 MPa,optionally for some applications selected from the range of 50 KPa to 10MPa. Useful elastic substrates include those having a Young's modulusless than or equal to 1 MPa, or less than or equal to 0.5 MPa, or lessthan or equal to 200 KPa. Useful elastic substrates include those havinga net bending stiffness selected from the range of 0.1×10⁴ GPa μm⁴ to1×10⁹ GPa μm⁴. In specific embodiments, the elastic substrate comprisesa material selected from the group consisting of: a polymer, aninorganic polymer, an organic polymer, a plastic, an elastomer, abiopolymer, a thermoset, a rubber silk and any combination of these. Inan embodiment, the elastic substrate comprises an organosiliconcompound, such as silicone. In an embodiment, the elastic substratecomprises PDMS (polydimethylsiloxane).

The invention provides systems and components for a range of deviceapplications, including optical, electronic, thermal, mechanical andchemical sensing and/or actuation. For example, in various embodiments,electronic circuits of this aspect comprise one or more of an energystorage device, a photonic device, an optical sensor, a strain sensor,an electrical sensor, a temperature sensor, a chemical sensor, anactuator, a communication device, a micro- or nano-fluidic device, anintegrated circuit or any component thereof. In a specific embodiment,an electronic circuit comprises a tissue mounted electronic device, aradio frequency antenna or a sensor compatible with magnetic resonanceimaging.

Appropriately geometrically designed and/or configured circuitembodiments of the invention provide the ability to use the electricalcircuits of the invention in unconventional situations and or for whereelectronic devices are typically unwanted, incompatible and/orinoperable. For example, electronic circuits of certain embodiments ofthe invention exhibit properties compatible with magnetic resonanceimaging. Such embodiments provide the ability to measure and/or controlproperties of materials, such as a tissue, interfacing with anelectronic circuit embodiment of the invention while magnetic resonanceimaging measurements are being prepared and/or made. In one aspect, forexample, the invention provides an electronic sensor compatible withmagnetic resonance imaging. A specific embodiment of this aspectcomprises: an elastic substrate; and a stretchable electrode arraysupported by the elastic substrate; the electrode array comprising aplurality of electrically conductive elements each having a primary unitcell shape, the electrically conductive elements connected in a sequencehaving a secondary shape providing an overall two-dimensional spatialgeometry characterized by a plurality of spatial frequencies; whereinthe stretchable electrode array is substantially transparent (e.g.,absorption less than 0.1, optionally for some applications less than0.05 and further optionally for some applications less than 0.01) toradio frequency electromagnetic radiation having frequencies selectedfrom the range of 1 MHz-100 GHz, and wherein the two-dimensional spatialgeometry of the stretchable electrode array allows for accommodation ofelastic strain along one or more in-plane or out of plane dimensions,thereby providing stretchability of the electronic sensor. In someembodiments, for example, the stretchable electrode array does notinclude any closed loop structures over an active area of the electronicsensor, such as no closed loops within the electrode pad. In certainembodiments, the stretchable electrode array does not result inobservable distortion or shadowing when using in combination withmagnetic resonance imaging. In embodiments, for example, the magneticresonance imaging includes exposure of the electronic sensor to RFpulses having the frequencies selected from the range of 100-300 MHz.

The invention includes stretchable circuits comprising a plurality ofelectronic devices or device components provided in an array, such asmultiple individually addressable and optionally electronically isolatedcircuit components or electronic devices. Use of array configurationsprovide the electronic circuits of the invention, in embodiments, withthe ability to interface with large areas or a surface, such asindividual distinct areas of the surface of a sample or tissue. In anaspect, the invention provides electrode arrays useful for interfacingwith a plurality of locations across an object or tissue. In a specificembodiment, an electrode array comprises: a plurality of stretchablemetallic or semiconducting device components supported by the elasticsubstrate, wherein each of the stretchable metallic or semiconductingdevice components independently comprises a plurality of electricallyconductive elements each having a primary unit cell shape, wherein theelectrically conductive elements of each stretchable metallic orsemiconducting device component are independently connected in asequence having a secondary shape providing an overall two-dimensionalspatial geometry characterized by a plurality of spatial frequencies;wherein the plurality of stretchable metallic or semiconducting devicecomponents provide a fill factor greater than or equal to 50% for anactive area of the electrode array; and wherein the two-dimensionalspatial geometries of the metallic or semiconducting device componentsallows for accommodation of elastic strain along one or more in-plane orout of plane dimensions, thereby providing stretchability of theelectrode array.

Embodiments of the invention also provide stretchable electronicdevices, such as devices incorporating electronic circuit embodiments ofthe invention. For example, incorporating electronic circuits comprisinga self-similar geometry and/or a two-dimensional spatial frequency intoa larger electronic device optionally results in providing theelectronic devices with beneficial features, such as enhancedstretchability. In embodiments, stretchable devices incorporate bothrigid and/or conventional circuits or circuit components andflexible/stretchable components in a configuration providing beneficialstretchability aspects to the overall configuration of the device. Forexample, in embodiments, the invention provides stretchable electronicdevices, such as electronic devices incorporating electronic circuitsand devices disclosed herein.

In an aspect, a stretchable electronic device comprises: a plurality ofrigid island structures supported by an elastic substrate; wherein eachof the rigid island structures independently comprises an inorganicsemiconductor device or device component; a plurality of stretchablemetallic or semiconducting device components electricallyinterconnecting at least a portion of the rigid island structures,wherein each of the stretchable metallic or semiconducting devicecomponents independently comprises a plurality of electricallyconductive elements each having a primary unit cell shape, wherein theelectrically conductive elements of each stretchable metallic orsemiconducting device component are independently connected in asequence having a secondary shape providing an overall two-dimensionalspatial geometry characterized by a plurality of spatial frequencies;wherein the two-dimensional spatial geometries of the metallic orsemiconducting device components allows for accommodation of elasticstrain along one or more in-plane or out of plane dimensions, therebyproviding stretchability of the stretchable electronic device.

Without wishing to be bound by any particular theory, there may bediscussion herein of beliefs or understandings of underlying principlesrelating to the devices and methods disclosed herein. It is recognizedthat regardless of the ultimate correctness of any mechanisticexplanation or hypothesis, an embodiment of the invention cannonetheless be operative and useful.

BRIEF DESCRIPTION OF THE DRAWINGS

FIGS. 1A, 1B and 1C provide examples of two-dimensional spatialgeometries useful with aspects of the invention.

FIG. 2 provides an overview of representative fractal-inspired layoutsfor hard-soft materials integration. Six different patterns of metalwires fully bonded to elastomer substrates demonstrate the applicationof deterministic fractal designs as general layouts for stretchableelectronics. These patterns include line, loop, and branch-likegeometries, and they are applicable to a broad range of active andpassive electronic components. In all cases, arc sections replace thesharp corners from the mathematically-defined fractal layouts to improvethe elastic mechanics (top row). FEM images of each structure underelastic tensile strain (middle row) and their corresponding experimentalMicroXCT images (bottom row) demonstrate the elastic mechanics. Thewires consist of layers of gold (300 nm) sandwiched by polyimide (1.2μm) and mounted on an elastomeric substrate (0.5 mm).

FIG. 3 provides an overview of an implementation of Peano curves forstretchable electrodes with fractal designs. Panel (a) provides anillustration of three iterations of a two-dimensional Peano curve. Nineversions of the (N−1)th curve (dotted red box) connect together with thesolid red lines to construct the Nth curve. Panel (b) depicts arcsections replace the sharp bends to enhance the mechanics. Panel (c)provides an image of metal wires with Peano layouts, with an overallgeometry that spell out the characters in “ILLINOIS”, mounted on skin.Here, each letter consists of a series of first and second order Peanocurves. Panel (d) provides a detailed section of the N (solid red box)from panel (c) in block diagram form (small blocks are first ordercurves, large blocks are second order curves) and as wires. Panel (e)provides an optical image and panel (f) provides a scanning electronmicroscopy image of third order Peano-based wires on skin and askin-replica (colorized metal wires), respectively, showing theconformal contact of the wires on the substrate. The experimentaldimensions are as follows: the 2^(nd) order gold wires have R=620 μm andw=70 μm; the 3^(rd) order gold wires have R=190 μm and w=70 μm; and thesilicon nanomembranes (Si NMs) have R=190 μm and w=70 μm.

FIGS. 4a-4c provide an overview of mechanical characterization offractal-based structures of metals and semiconductors. FIG. 4a and FIG.4d depict optical images of metal wires patterned with second and thirdorder half-and-half Peano layouts. The samples connect to a stretchableset of four wires used for four point probing. FIG. 4b and FIG. 4eprovide plots of differential resistances measured at increasing levelsof maximum applied strain. The onset of plastic deformation correspondsto the strain at which the differential resistance is non-zero. FIG. 4cand FIG. 4f illustrate a comparison between experimental optical imagesand FEM-constructed images of the second and third order structures fordifferent levels of strain. FIG. 4g and FIG. 4h provide MicroXCT and FEMimages of Si NMs patterned into Peano layouts and fully bonded to a 40%prestrained elastomeric substrate. Upon release and uniaxial stretching,both structures dissipate the mechanical stress via microscale buckles,which is indicative of a mechanical regime well suited for high levelsof elastic deformation.

FIGS. 5a-5f provide an overview of fractal-based epidermal devices whichillustrate how a patch type device integrates the recording, referenceand ground electrodes. FIG. 5a provides a schematic image of amultifunctional device based on a Greek cross fractal design. FIG. 5bprovides a corresponding image of the device on a wafer, and FIG. 5cprovides an image of the device mounted on skin. FIG. 5d depicts thedevice used in a Joule heating operation mode by driving currentsthrough the insulated heater wires, FIG. 5e provides data for fromdevice used for temperature sensing via four point probing, and FIG. 5fprovides data from the device used for ECG measurements.

FIG. 6a-6d provides an overview of the RF properties of stretchablefractal structures. FIG. 6a provides data illustrating return lossparameters of a box fractal antenna under different amounts of tensilestrain. The inset in FIG. 6a shows an optical image of an unstrainedantenna fully bonded onto an elastomer. FIG. 6b provides an opticalimage of a box fractal antenna under 30% tensile strain and acorresponding FEM-constructed image of the same structure. FIG. 6cprovides far-field profiles of the antenna under no strain and 30%strain. FIG. 6d depicts a cross sectional MRI image of different coppersamples (labeled 1-7) attached to a bottle of water. The solid and meshsamples display shadowing in the MRI image while the fractal-basedsamples display no such shadowing. The sample layouts all are 1×1 inchesand are: (1) solid square, (2) thick mesh, (3) thin mesh, (4) horizontaland vertical stripes separated by a spacer, (5) Peano curve, (6) Hilbertcurve, (7) Serpentine Greek cross.

FIG. 7 provides an overview of the third order Peano curve, as depictedin FIG. 1a , contains spring-like motifs at small (left), medium(middle, green highlight), and large (right, green highlight) lengthscales. Additionally, the spring-like motifs can orient along the x- ory-axes, as illustrated by the two medium springs in the middle plot.

FIG. 8 provides an overview of iteration processes. Panel (a)illustrates nine versions of the (N−1)th sub-unit cell link togetherinto a continuous line to construct the Nth iteration is constructed bylinking together. The arrows denote the starting and ending point ofeach of the sub-unit cells. Each of these nine subunits can orienteither vertically or horizontally without changing its starting andending point, as demonstrated with the first order iterative subunit(Panel (b)). As such, there exist many variations of the Peano curve(272 different second order Peano curves alone). Panel (c) shows thesimulated and experimental structures consist of a series of arcsections with solid angles of 90, 180, and 270 degrees. The intersectionpoints between arc sections coincide with the mathematically-definedPeano curve, which constrains their position. As such, the radii ofcurvature (R), wire width (w), and wire thickness (t) define theabsolute geometry for a given layout type and fractal dimension.

FIG. 9 provides an overview of the experimental system and results.Panel (a) shows the experimental setup for device calibration andtesting. To measure the device resistance via four point probe, wesupply 1 mA of constant current to the device with one pair of wires andmeasure the voltage drop in the device with the second pair of wiresusing a lock-in amplifier. Panel (b) provides a detailed picture of thefractal-based sample bonded to ACF cables and mounted on the uniaxialstretcher. Panel (c) provides a detailed picture of the prestrainedfractal sample during mechanical loading and unloading. Panel (d)provides a summary of the calculated and experimentally-measuredstretchabilities for half-and-half samples along the x- and y-axes,delineating the transition point from elastic to plastic deformation.

FIG. 10 provides an overview of stretched devices. Panel (a) illustratesa comparison between experimental optical images and FEM-constructedimages of an all-vertical second order structure under no strain,pre-strain, and stretching. Panel (b) provides a plot of differentialresistances measured in a sample for increasing amounts of maximumapplied strain.

FIG. 11 provides a plot of silicon nanomembrane resistance as a functionof tensile strain for two different half-and-half samples (black and redlines). Failure is clearly defined when the membranes crack and theresistance dramatically increases. The maximum elastic strains here areconsistent with the simulated FEM strain of 63%.

FIG. 12a provides data showing calculated stretchability of anall-vertical Peano silicon nanomembrane strained along the y-axis fordifferent membrane thicknesses. The planar geometry is identical to thatpresented in FIG. 3. FIG. 12b provides a schematic illustration showingthat as the membrane thickness increases, the strained membranesexperience mechanical regimes spanning wrinkling, micro-buckling, andglobal buckling. The maximum device stretchability coincides with themicro-buckle regime, indicating that micro-buckles are particularly wellsuited for dissipating stress in the mechanically hard silicon membranesmounted on a soft substrate.

FIG. 13 shows an optical image of a skin-mounted third order Peanotemperature sensor in Panel (a), with simultaneous measurements oftemperature on skin with an infrared camera showing good agreement(Panel (b)). Panel (c) shows a third order Peano device functions as aheater by passing larger amounts of current (order of 10 mA) through thegold wires. The heat distribution within the device is uniform with andwithout stretching due to the space filling properties of the wirelayout.

FIG. 14 shows space filling features of a cross. Panel (a) shows thefirst order structure is a cross. Higher order structures areiteratively constructed by adding crosses of exponentially smallerdimension, as shown. Panel (b) shows that, for the electrodes featuredhere, the connectivity between the structures (red lines) is definedsuch that there are no closed loops of wire in the network. Panel (c)depicts where the straight lines in the mathematical structure arereplaced with serpentine layout to enable stretchability.

FIG. 15 provides data showing a comparison of EMG signals taken fromfractal epidermal electrodes (Panel (a)) and gel-based electrodes (Panel(b)) mounted on the forearm show comparable signal-to-noise ratios. Thepeaks are due to the clenching of the fist.

FIG. 16 provides an illustration of the first three iterations of theVicsek fractal.

FIG. 17a depicts a box fractal antenna geometry. The feed point of theantenna is chosen so that the impedance of its fundamental modesufficiently matches with the coaxial feed. FIG. 17b provides datashowing return loss spectra of the box fractal antenna for threedifferent levels of stretching. It is noted that the bandwidth of thebox fractal is very narrow, due to the large inductive load built intothe antenna. FIG. 17c illustrates a far field intensity profile of theunstretched antenna at resonance.

FIG. 18 provides an overview of elastic mechanics of five differentPeano-based wire structures. Calculated stretchability of metal wiresmounted on an elastomer in five different second order Peano layouts,given a maximum principal strain criterion of 0.3% in any section of thewires. The layouts range from “all-horizontal” (subunits are alloriented along the x-axis) to “all-vertical” (subunits are all orientedalong the y-axis). The strain criterion defines the transition fromelastic to plastic deformation in the wires.

FIG. 19 provides an overview of simulated uniaxial elasticstretchability for serpentine wires as a function of arc solid angle.The inset of the middle column defines the arc solid angle. Thecross-sectional geometries and materials match those from FIG. 3, andall structures have R=620 μm and w=70 μm (defined in Figure S2). Thesesimulations clearly display that elastic stretchability increases as afunction of arc angle in these primitive serpentine geometries. As such,deterministically defining the arc section geometries in wire-typestructures can help optimize the mechanics.

FIG. 20 provides an overview of simulated biaxial stretchability as afunction of unit cell size for half-and-half Peano structures. Thecross-sectional geometries and materials match those from FIGS. 4a-4h ,and all structures have w=70 μm (defined in FIG. 8). The structures withunit cell sizes between 1.5 mm and 4.5 mm display biaxialstretchabilities greater than 20% and are compatible with the elasticproperties of skin.

FIGS. 21A-21E provide an overview of aspects in battery layout anddesign. FIG. 21A provides a schematic illustration of a completeddevice, in a state of stretching and bending. FIG. 21B provides anexploded view layout of the various layers in the battery structure.FIG. 21C provides an illustration of ‘self-similar’ serpentinegeometries used for the interconnects (black: 1^(st) level serpentine;yellow: 2^(nd) level serpentine). FIG. 21D shows optical images of theAl electrode pads and self-similar interconnects on Si wafer (leftpanel; top down view; ˜4 unit cells), after transfer printing on a sheetof silicone (middle panel; top down view, in a bent geometry), and withmoulded slurries of LiCoO₂ (right panel; top down view, in a bentgeometry). FIG. 21E shows optical images of the Cu electrode pads andself-similar interconnects on Si wafer (left panel; top down view; ˜4unit cells), after transfer printing on a sheet of silicone (middlepanel; top down view, in a bent geometry), and with moulded slurries ofLi₄Ti₅O₁₂ (right panel; top down view, in a bent geometry). Scale barsin FIG. 21D and FIG. 21E are 2 mm.

FIG. 22 provides an overview of experimental and computational studiesof buckling physics in interconnects with self-similar serpentinelayouts. Optical images and corresponding finite element analysis (FEA)of symmetric (left column) and anti-symmetric (middle column)deformation modes, for various levels of applied tensile strain (ε). Thecolor in the FEA results represents the maximum principal strains of themetal layer. The scale bar is 2 mm. The right column shows theinterconnect structures after releasing the applied strain.

FIGS. 23A-23H provide an overview of electrochemical and mechanicalproperties of the battery. FIG. 23A provides results of Galvanostaticcharging and discharging of the battery electrodes without (black) andwith 300% uniaxial strain (red). FIG. 23B provides results showingcapacity retention (black square) and coulombic efficiency (red circle)over 20 cycles with a cutoff voltage of 2.5-1.6 V. FIG. 23C providesdata showing output power as a function of applied biaxial strain. FIG.23D shows an image of operation of a battery connected to a red LEDwhile FIGS. 23E-23H show an image of a device biaxially stretched to300% (FIG. 23E), folded (FIG. 23F), twisted (FIG. 23G), and compliantwhen mounted on the human elbow (FIG. 23H).

FIGS. 24A-24D provide an overview of a stretchable system for wirelesscharging. FIG. 24A provides a circuit diagram. FIG. 24B shows an imageof the integrated system with different components labeled. FIG. 24Cprovides data showing characterization of the wireless coil with analternating voltage input at a frequency of 44.5 MHz (black) and theresulting direct voltage output (red), as indicated in FIG. 24A. FIG.24D provides data showing charging voltage (top) and current (bottom)curves as a stretchable battery is charged with 3 V output from thewireless circuit. The scale bar in FIG. 24B is 1 cm.

FIG. 25 provides an illustration of the dimensions for the self-similarinterconnect (copper layer).

FIG. 26 provides a schematic illustration of the fabrication process,and images of the moulded cathode (top right) and anode slurry (bottomright) on water soluble tape.

FIG. 27 provides an SEM image of a buckled Al foil (600 nm)/PI (1200 nm)bilayer on the surface of a sheet of ecoflex after releasing a prestrainof ˜30%. This bilayer structure resembles the types of laminatedAl/polymer packaging materials that are used in pouch cells, to blockthe permeation of water, air and solvent.

FIGS. 28A and 28B provide deformed configurations (FEA results) of theself-similar electrode for symmetric (FIG. 28A) and anti-symmetric (FIG.28B) buckling modes under an applied strain of 50%, from differentviewing angles (i.e., top, front, side, and three-dimensional (3D)views).

FIG. 29 illustrates the maximum value (ε_(max)) of the maximum principalstrain in the metal layer of the self-similar interconnect as a functionof the applied strain (ε_(appl)), together with the evolution of thedeformations.

FIG. 30 illustrates the distribution of maximum principal strain in themetal layer when its maximum value reaches 1%; (top) the 2-orderself-similar interconnect; and (bottom) the 1-order interconnect. Thetwo structures have the same overall dimensions, and cross-sections.

FIG. 31 provides a plot showing the maximum value (ε_(max)) of themaximum principal strain in the metal layer of the interconnect as afunction of the applied strain (ε_(appl)), for the self-similar andsimple serpentine designs. The two interconnects have the same totallength (l_(total)), span (L), amplitude (h), width (w), and thickness(t).

FIG. 32 provides the results of finite element analyses of the bucklingprofiles of a vertically aligned self-similar interconnect undercompression, and its comparison with optical images from experiment. Thecolor contours in the FEA results represent the distribution of maximumprincipal strain in the metal layer.

FIG. 33A shows the layout of Al and Cu pads, and FIG. 33B illustratesthe dependences of fill factor on the size of a representative unit celland the radius of the Al pad. The offset distance (d) is set to be 0.5mm in the model to avoid possible short circuits as the battery isstretched.

FIG. 34 provides a Nyquist impedance plot for the pouch type stretchablebattery from 1 MHz to 10 mHz with an a.c. perturbation amplitude of 10mV.

FIG. 35 provides data showing open circuit voltage decay curves (top)and leakage current curves (bottom) for batteries in variousconfigurations, measured at room temperature.

FIG. 36 illustrates capacity retention (black squares) and coulombicefficiency (red circles) over 20 cycles with a cutoff voltage of 2.5-1.6V for coin cell geometries with exactly the same slurries and thecapacitance matched cathode and anode geometries.

FIG. 37 illustrates capacity retention curves with depth of discharge of100% (red circle curve, cut-off voltage of 1.60-2.50 V) and ˜75% (blacksquare curve, cut-off voltage of 2.25-2.50 V).

FIG. 38 provides I-V curve of a commercial red light emitting diode,showing its turn on voltage at around 1.7 V.

FIG. 39 provides a schematic illustration of the layout of a wirelessrecharging system (top); calculated deformation and distribution ofmaximum principal strain under an applied strain of 32.3%, for arepresentative component of the wireless coil, with both the discretediode and serpentine interconnect (middle); calculated deformation anddistribution of maximum principal strain under an applied strain of32.6%, for a representative component of the wireless coil, with onlythe serpentine interconnect (bottom).

FIG. 40A depicts a calculated distribution of maximum principal strainin the whole structure, and FIG. 40B depicts a calculated distributionof substrate normal strain (ε₃₃) at the diode/substrate interface, whenthe system is stretched by 30% along the vertical direction.

FIG. 41 depicts the input and output electrical characteristics of thewireless charging systems. The Schottky diode rectifies the alternatingvoltage input from the functional generator (pink curve), to yield arectified output (blue curve), which oscillates nominally from 0 V to4.6 V. The parallel 1.7 nF capacitor integrates this oscillation to givea current source with a behavior closer to direct current (red curve).Increasing the capacitance (e.g. 17 nF) further smooths the current(black curve).

FIG. 42 provides I-V curve of the wireless coil with the rectifyingchip, showing its series resistance of about 2.3 KΩ.

FIG. 43A provides input and output voltages (black) and currents (red)of the wireless coil 18 μm thick copper foil. FIG. 43B provides opticalmicrographs of a 7 μm thick Cu coil at different levels of applieduniaxial strain. The scale bars are all 1 cm. FIG. 43C provides finiteelement analysis of stretching of a segment of a serpentine coil with athickness of 7 μm.

FIGS. 44A and 44B provide a comparison between the measured currentchange (FIG. 44A) and the simulated result (FIG. 44B) in the wirelesscoil charging circuit.

FIG. 45. (a) Optical images of the Al electrode pads and self-similarinterconnects on Si wafer (left panel; top down view; ˜4 unit cells),after transfer printing on a sheet of silicone (middle panel; obliqueview, in a bent geometry), and with moulded slurries of LiCoO₂ (rightpanel; oblique view, in a bent geometry), for a stretchable Li-ionbattery; (b) Schematic illustration on the geometric construction of theself-similar rectangular interconnect; (c) Schematic illustration on thegeometric construction of the self-similar serpentine interconnect. Thescale bars in (a) are 2 mm. (a) is reprinted with permission from Xu etal. [19], Copyright 2013, Nature Publishing Group.

FIG. 46. (a) A freely suspended 1^(st) order rectangular interconnect,clamped at the left end, and subject to an axial force N, shear force Q,and bending moment M, at the right end. (b) Exploded view and free bodydiagram of the k^(th) unit cell of 1^(st) order rectangularinterconnect.

FIG. 47. The exploded view of a representative unit cell for the (a)2^(nd) order and (b) 3^(rd) order self-similar rectangular interconnect.

FIG. 48. Schematic illustration on the geometric construction of a3^(rd) order generalized self-similar serpentine interconnect.

FIG. 49. The effect of self-similar order on the flexibility: Thedimensionless flexibility components (T ₁₁ ^((n)), T ₁₂ ^((n)) and T ₃₃^((n)) (a) and (T ₂₂ ^((n)) and T ₂₃ ^((n))) (b) versus the self-similarorder. In the FEA, the width is fixed as w=0.4 l⁽¹⁾ for the structuresof different orders.

FIG. 50. The dimensionless stretchability versus the height/spacingratio (η⁽¹⁾) for different number (m⁽¹⁾) of unit cells for the 1^(st)order serpentine interconnect.

FIG. 51. The dimensionless stretchability of the 2^(nd) order serpentineinterconnect versus the height/spacing ratio (η⁽²⁾) for different number(m⁽²⁾) of unit cells, with (m⁽¹⁾, η⁽¹⁾)=(8,2).

FIG. 52. The dimensionless stretchability as a function of theself-similar order. In the FEA, the width is fixed as w=0.4 l⁽¹⁾ for thestructures of different orders.

FIG. 53. Design optimization of the 2^(nd) order serpentineinterconnects for island-bridge structure. (a) Schematic of theisland-bridge structure with a 8×8 array, and illustration on thegeometric parameters; (b) The maximum stretchability versus the number(m⁽²⁾) of unit cells (left panel), and the optimized configuration(right panel).

FIG. 54. (a) Optical images of electrode pads and fractal inspiredinterconnects on a silicon wafer (top panel; top down view; ˜4 unitcells), after transfer printing on a sheet of silicone (middle panel;oblique view, in a bent geometry), and with moulded slurries of LiCoO₂(bottom panel; oblique view, in a bent geometry), for a stretchableLi-ion battery; (b) Optical images and corresponding conventional FEAresults of symmetric deformation modes, for various levels of appliedtensile strain s. The scale bars in (a) and (b) are 2 mm. (a) and (b)are reprinted with permission from Xu et al. (2013), Copyright 2013,Nature Publishing Group.

FIG. 55. Schematic illustration of the geometric construction of afractal inspired interconnect.

FIG. 56. Schematic illustration of the equivalent structure for afractal inspired interconnect. (a) An order-n fractal interconnectcomposed of vertically aligned order-(n−1) interconnects, andhorizontally aligned order-(n−2) interconnects; (b) Equivalent structureconsisting of only straight beams.

FIG. 57. Illustration of the hierarchal computational model (HCM) for a2^(nd) order fractal interconnect. (a) Stage I—unraveling the 2^(nd)order structure, in which the entire interconnect is modeled by anequivalent structure of straight beams; (b) Stage II—unraveling each1^(st) order structure, studied using the original geometry of thestructure.

FIG. 58. Elastic stretchability versus the order for fractalinterconnects from n=1 to 4, with (m, η)=(4,8√{square root over (11)}),the thickness/width aspect ratio (t/w=0.03), and the width to spacingratio (w/l⁽¹⁾=0.4), for structures of different fractal orders.

FIG. 59. The maximum principal strain versus the applied strain for a1^(st) order serpentine interconnect with (m, η)=(4,8/√{square root over(11)}). The inset illustrates the deformed pattern of the serpentineinterconnect as the elastic stretchability is reached.

FIG. 60. (a) The maximum principal strain versus the applied strain fora 2^(nd) order fractal interconnect with (m, η)=(4,8/√{square root over(11)}); (b) the undeformed and deformed configurations when the 2^(nd)order structure is fully unraveled.

FIG. 61. (a) The maximum principal strain versus the applied strain fora 3^(rd) order fractal interconnect with (m, η)=(4,8/√{square root over(11)}); (b) the undeformed and deformed configurations when the 3^(rd)and 2^(nd) order structures are fully unraveled.

FIG. 62. (a) The maximum principal strain versus the applied strain fora 4^(th) order fractal interconnect with (m, η)=(4,8/√{square root over(11)}); (b) the undeformed and deformed configurations when the 4^(th),3^(rd) and 2^(nd) order structures are fully unraveled.

FIG. 63. Symmetric (left panel) and anti-symmetric (right panel)deformation modes from experiments (Xu et al., 2013) and numericalresults by the HCM, for various levels of applied tensile strain(0300%). The scale bar is 2 mm.

FIG. 64. The maximum principal strain in the metal layer versus theapplied strain from conventional FEA and the HCM, for the fractalinterconnects adopted in the experiment of Xu et al. (2013).

FIG. 65. The 1^(st) (a) and 2^(nd) (b) order fractal interconnects, withthe same total length (16.77 mm) of interconnect, spacing (1.6 mm)between the device islands, height (0.4 mm) of the 1^(st) orderinterconnect, width (w=30 μm), and thickness (t=3.0 μm).

FIG. 66. The free body diagram of the 1st order rectangular (a) andserpentine (b) interconnect, after separating the structure at thecenter. The interconnects are clamped at two ends, and subjected to adisplacement loading. The red sites in the figure schematicallyillustrate the position of maximum strain.

FIG. 67. The non-dimensional stretchabilities of various geometricparameters, as calculated by analytic model, FEA based on infinitesimaldeformation and FEA based on finite deformation: (a) 1st orderserpentine interconnect; (b) 2nd order serpentine interconnect, withm(1)=8 and η(1)=2; (c) 2nd order serpentine interconnect, with m(2)=1and η(2)=2. In the FEA of both 1st and 2nd order structures, the widthis fixed as a typical value of w=0.4 l(1).

DETAILED DESCRIPTION

In general, the terms and phrases used herein have their art-recognizedmeaning, which can be found by reference to standard texts, journalreferences and contexts known to those skilled in the art. The followingdefinitions are provided to clarify their specific use in the context ofthe invention.

“Two-dimensional spatial geometry” refers to an arrangement ofmaterials, structures or components in space such that they aredistributed along two independent axes, such as two axes defining aplane, or across a surface of an object or substrate. In embodiments, anobject possessing a two-dimensional spatial geometry includes materials,structures or components traversing, at least in part, along lengths ofeach of two dimensions, such as in an x-y plane. As used herein“two-dimensional spatial geometry” is distinguished from aone-dimensional geometry, such as the geometry corresponding to a lineor a thin film of a material extending directly between two points inspace. In embodiments, a two-dimensional spatial geometry comprises anarrangement of materials, structures or components in a conformal manneracross a surface, for example, a planar or non-planar surface.

A “two-dimensional spatial geometry characterized by a plurality ofspatial frequencies” refers to an arrangement of materials, structuresor components in space such that they are distributed along twoindependent axes and where the materials, structures or componentsthemselves are characterized by periodic or repeating spatialconfigurations characterized by at least two different length scales. Inembodiments, the plurality of spatial frequencies are characterized by aplurality of length scales, such as a first order, unit-cell or shortrange length scale, a second order, secondary shape or longer rangelength scale, a third order, tertiary shape or long range length scale,etc. For some embodiments, the plurality of spatial frequencies of atwo-dimensional spatial geometry can be characterized by a frequencyanalysis of the spatial distribution of materials arranged in thetwo-dimensional spatial geometry, such as a Fourier transform analysis,yielding two or more maxima characterizing the spatial distribution ofmaterials. In embodiments, two-dimensional spatial geometriescharacterized by a plurality of spatial frequencies include, but are notlimited to, self-similar geometries, fractal like geometries, geometrieshaving a fractal dimension of between 1 and 2, optionally for someapplications a fractal dimension of between 1.3 and 2. In embodiments,two-dimensional spatial geometries characterized by a plurality ofspatial frequencies include, but are not limited to,spring-within-a-spring type geometries.

“Fractal-based two-dimensional geometry” refers to a two-dimensionalgeometry of the stretchable metallic or semiconducting device componentthat is based-on or otherwise derived from one or more deterministicfractal patterns. Useful fractal patterns for fractal-based geometriesof the invention include, but are not limited to, Peano, Vicsek, GreekCross, and Hilbert fractals. The spatial layouts of fractal-basedgeometries incorporate features of one or more fractal patterns, forexample, using the lines, perimeters or shapes (in part or in whole), asa layout design rule(s). Fractal-based geometries may optionallyincorporate modifications of a fractal pattern to enhance overallmechanical or other physical properties, such as stretchability and/orfill factor, for example, by replacing sharp corners in fractal patternswith loops or straight lines in fractal patterns with serpentinegeometries. In an embodiment, fractal-based geometries useful in thepresent invention are characterized by an approximate fractal dimensionselected from the range of 1.2 to 2. In an embodiment, fractal-basedgeometries useful in the present invention have spatial propertiesdefined by precise or approximate iterative rules, which characterizehigher order fractal-like patterns by multiple length scales. Forexample, if the Nth iterative pattern has a length scale L_N and theN−1th iterative pattern has a length sale L_N−1, then average ratioL_N/L_N−1 for all N's is A, and the ratio L_N/L_N−1 for a given N falls±30% of A. Fractal-based geometries for some embodiments combine two ormore basic fractal-like patterns into lines or meshes to create largerfractal-like pattern (see, e.g., FIG. 3C).

“Fill factor” refers to the percentage of an area between two elements,such as first and second electrical contact points, device islands orcontact pads, that supports and/or is occupied by a material, elementand/or device component. In an embodiment, for example, fill factorrefers to the percentage of a region of a surface, such as a surface ofan elastic substrate or layer provided thereon, that supports (andoptionally is in physical contact with) one or more stretchable metallicor semiconducting device components. In an embodiment, for example, fillfactor refers to the percentage of a region of an active area of adevice that supports (and optionally is in physical contact with) one ormore stretchable metallic or semiconducting device components. In anembodiment, for example, fill factor refers to the percentage of asurface extending between two device islands, such as semiconductordevice components or devices, that supports (and optionally is inphysical contact with) one or more stretchable metallic orsemiconducting device components. In an embodiment, for example, fillfactor refers to the percentage of the area of the portion of a surfaceextending between two elements, such as first and second electricalcontact points or contact pads, that supports one or more stretchablemetallic or semiconducting device components.

“Functional layer” refers to a layer that imparts some functionality tothe device. For example, the functional layer may contain semiconductorcomponents. Alternatively, the functional layer may comprise multiplelayers, such as multiple semiconductor layers separated by supportlayers. The functional layer may comprise a plurality of patternedelements, such as interconnects running between or below electrodes orislands. The functional layer may be homogeneous or may have one or moreproperties or materials that are inhomogeneous. “Inhomogeneous property”refers to a physical parameter that can spatially vary, therebyeffecting the position of the neutral mechanical plane within amultilayer device.

“Structural layer” refers to a layer that imparts structuralfunctionality, for example by supporting and/or encapsulating and/orpartitioning device components.

“Semiconductor” refers to any material that is an insulator at a verylow temperature, but which has an appreciable electrical conductivity ata temperature of about 300 Kelvin. In the present description, use ofthe term semiconductor is intended to be consistent with use of thisterm in the art of microelectronics and electronic devices. Usefulsemiconductors include those comprising elemental semiconductors, suchas silicon, germanium and diamond, and compound semiconductors, such asgroup IV compound semiconductors such as SiC and SiGe, group III-Vsemiconductors such as AlSb, AlAs, AlN, AlP, BN, BP, BAs, GaSb, GaAs,GaN, GaP, InSb, InAs, InN, and InP, group III-V ternary semiconductorssuch as Al_(x)Ga_(1-x)As, group II-VI semiconductors such as CsSe, CdS,CdTe, ZnO, ZnSe, ZnS, and ZnTe, group I-VII semiconductors such as CuCl,group IV-VI semiconductors such as PbS, PbTe, and SnS, layersemiconductors such as PbI₂, MoS₂, and GaSe, and oxide semiconductorssuch as CuO and Cu₂O. The term semiconductor includes intrinsicsemiconductors and extrinsic semiconductors that are doped with one ormore selected materials, including semiconductors having p-type dopingmaterials and n-type doping materials, to provide beneficial electronicproperties useful for a given application or device. The termsemiconductor includes composite materials comprising a mixture ofsemiconductors and/or dopants. Specific semiconductor materials usefulfor some embodiments include, but are not limited to, Si, Ge, Se,diamond, fullerenes, SiC, SiGe, SiO, SiO₂, SiN, AlSb, AIAs, AlIn, AlN,AlP, AlS, BN, BP, BAs, As₂S₃, GaSb, GaAs, GaN, GaP, GaSe, InSb, InAs,InN, InP, CsSe, CdS, CdSe, CdTe, Cd₃P₂, Cd₃As₂, Cd₃Sb₂, ZnO, ZnSe, ZnS,ZnTe, Zn₃P₂, Zn₃As₂, Zn₃Sb₂, ZnSiP₂, CuCl, PbS, PbSe, PbTe, FeO, FeS₂,NiO, EuO, EuS, PtSi, TIBr, CrBr₃, SnS, SnTe, PbI₂, MoS₂, GaSe, CuO,Cu₂O, HgS, HgSe, HgTe, HgI₂, MgS, MgSe, MgTe, CaS, CaSe, SrS, SrTe, BaS,BaSe, BaTe, SnO₂, TiO, TiO₂, Bi₂S₃, Bi₂O₃, Bi₂Te₃, BiI_(a), UO₂, UO₃,AgGaS₂, PbMnTe, BaTiO₃, SrTiO₃, LiNbO₃, La₂CuO₄, La_(0.7)Ca_(0.3)MnO₃,CdZnTe, CdMnTe, CuInSe₂, copper indium gallium selenide (CIGS), HgCdTe,HgZnTe, HgZnSe, PbSnTe, Tl₂SnTe₅, Tl₂GeTe₅, AlGaAs, AlGaN, AlGaP,AlInAs, AlInSb, AlInP, AlInAsP, AlGaAsN, GaAsP, GaAsN, GaMnAs, GaAsSbN,GaInAs, GaInP, AlGaAsSb, AlGaAsP, AlGaInP, GaInAsP, InGaAs, InGaP,InGaN, InAsSb, InGaSb, InMnAs, InGaAsP, InGaAsN, InAlAsN, GaInNAsSb,GaInAsSbP, and any combination of these. Porous silicon semiconductormaterials are useful for aspects described herein. Impurities ofsemiconductor materials are atoms, elements, ions and/or molecules otherthan the semiconductor material(s) themselves or any dopants provided tothe semiconductor material. Impurities are undesirable materials presentin semiconductor materials which may negatively impact the electronicproperties of semiconductor materials, and include but are not limitedto oxygen, carbon, and metals including heavy metals. Heavy metalimpurities include, but are not limited to, the group of elementsbetween copper and lead on the periodic table, calcium, sodium, and allions, compounds and/or complexes thereof.

A “semiconductor component” broadly refers to any semiconductormaterial, composition or structure, and expressly includes high qualitysingle crystalline and polycrystalline semiconductors, semiconductormaterials fabricated via high temperature processing, dopedsemiconductor materials, inorganic semiconductors, and compositesemiconductor materials. In some embodiments, for example, asemiconductor component is a semiconductor device or component thereof.

A “component” is used broadly to refer to an individual part of adevice. An “interconnect” is one example of a component, and refers toan electrically conducting structure capable of establishing anelectrical connection with another component or between components. Inparticular, an interconnect may establish electrical contact betweencomponents that are separate. Depending on the desired devicespecifications, operation, and application, an interconnect is made froma suitable material. Suitable conductive materials includesemiconductors. In some embodiments, for example, a component is acomponent of a semiconductor device.

Other components include, but are not limited to, thin film transistors(TFTs), transistors, electrodes, integrated circuits, circuit elements,control elements, microprocessors, transducers, islands, bridges andcombinations thereof. Components may be connected to one or more contactpads as known in the art, such as by metal evaporation, wire bonding,and application of solids or conductive pastes, for example.

“Neutral mechanical plane” (NMP) refers to an imaginary plane existingin the lateral, b, and longitudinal, l, directions of a device. The NMPis less susceptible to bending stress than other planes of the devicethat lie at more extreme positions along the vertical, h, axis of thedevice and/or within more bendable layers of the device. Thus, theposition of the NMP is determined by both the thickness of the deviceand the materials forming the layer(s) of the device.

“Coincident” refers to the relative position of two or more objects,planes or surfaces, for example a surface such as a neutral mechanicalplane that is positioned within or is adjacent to a layer, such as afunctional layer, substrate layer, or other layer. In an embodiment, aneutral mechanical plane is positioned to correspond to the moststrain-sensitive layer or material within the layer.

“Proximate” refers to the relative position of two or more objects,planes or surfaces, for example a neutral mechanical plane that closelyfollows the position of a layer, such as a functional layer, substratelayer, or other layer while still providing desired conformabilitywithout an adverse impact on the strain-sensitive material physicalproperties. “Strain-sensitive” refers to a material that fractures or isotherwise impaired in response to a relatively low level of strain. Ingeneral, a layer having a high strain sensitivity, and consequentlybeing prone to being the first layer to fracture, is located in thefunctional layer, such as a functional layer containing a relativelybrittle semiconductor or other strain-sensitive device element. Aneutral mechanical plane that is proximate to a layer need not beconstrained within that layer, but may be positioned proximate orsufficiently near to provide a functional benefit of reducing the strainon the strain-sensitive device element.

“Unitary” refers to an object formed as a single piece or undividedwhole.

The terms “direct and indirect” describe the actions or physicalpositions of one component relative to another component, or one devicerelative to another device. For example, a component that “directly”acts upon or touches another component does so without intervention froman intermediary. Contrarily, a component that “indirectly” acts upon ortouches another component does so through an intermediary (e.g., a thirdcomponent).

“Electronic device” generally refers to a device incorporating aplurality of components, and includes large area electronics, printedwire boards, integrated circuits, component arrays, electrophysiologicaland/or biological and/or chemical sensors, and physical sensors (e.g.,temperature, acceleration, etc.).

“Sensing” refers to detecting the presence, absence, amount, magnitudeor intensity of a physical and/or chemical property. Useful electronicdevice components for sensing include, but are not limited to electrodeelements, chemical or biological sensor elements, pH sensors,accelerometers, temperature sensors and capacitive sensors.

“Island” refers to a relatively rigid component of an electronic devicecomprising a plurality of semiconductor components. “Bridge” refers tostructures interconnecting two or more islands or one island to anothercomponent. The invention includes electronic devices having bridgestructures comprising electrical interconnects, such as stretchableelectrical interconnects provided between and in electrical contact withsemiconductor device components.

“Encapsulate” refers to the orientation of one structure such that it isat least partially, and in some cases completely, surrounded by one ormore other structures. “Partially encapsulated” refers to theorientation of one structure such that it is partially surrounded by oneor more other structures, for example, wherein 30%, or optionally 50% oroptionally 90%, of the external surfaces of the structure is surroundedby one or more structures. “Completely encapsulated” refers to theorientation of one structure such that it is completely surrounded byone or more other structures.

“Contiguous” refers to materials or layers that are touching orconnected throughout in an unbroken sequence. In one embodiment, acontiguous layer of a device has not been manipulated to remove asubstantial portion (e.g., 10% or more) of the originally providedmaterial or layer.

“Active circuit” and “active circuitry” refer to one or more componentsconfigured for performing a specific function. Useful active circuitsinclude, but are not limited to, amplifier circuits, multiplexingcircuits, current limiting circuits, integrated circuits, impedancematching circuits, wireless power harvesting circuits, wireless datatransmission circuits, transistors and transistor arrays.

“Substrate” refers to a material, layer or other structure having asurface, such as a receiving surface or supporting surface, that iscapable of supporting one or more components or electronic devices. Acomponent that is “bonded” to the substrate refers to a component thatis in physical contact with the substrate and unable to substantiallymove relative to the substrate surface to which it is bonded. Unbondedcomponents or portions of a component, in contrast, are capable ofsubstantial movement relative to the substrate. In an embodiment, theinvention includes electronic devices having one or more free standingsemiconductor device components supported by a substrate, optionally inphysical contact with the substrate or in physical contact with one ormore intermediate structures supported by the substrate. In anembodiment, the invention includes electronic devices having one or moretethered semiconductor device components supported by, or optionallybonded to, one or more structures, such as a pedestal or array ofpedestals, independently connecting the semiconductor device componentsto the substrate.

“Free standing” refers to a configuration wherein a device or devicecomponent is supported by, but not bonded to, a substrate orintermediate structure provided between the device or device componentand the substrate. In an embodiment, for example, a substrate is able tomove relative to a free standing device or component supported by thesubstrate. In an embodiment, for example, a free standing device orcomponent is able to move relative to a substrate supporting the freestanding device or component. In some embodiments, for example, a freestanding configuration of a device or device component decouplesmovement and/or deformation of the substrate from the device or devicecomponent. In some embodiments, for example, a free standingconfiguration of a device or device component decouples forces generatedby elongation, compression or deformation of the substrate from thedevice or device component. In some embodiments, a free standing deviceor component is characterized by undergoing an associative interactionwith a substrate surface or intermediate structure provided thereon,such as a Van der Waals interaction, dipole-dipole interaction or othernon-covalent associative interaction. In an embodiment, a free standingdevice or component is not covalently bonded to the supporting surfaceof a substrate.

“Tethered” refers to a configuration wherein a device or component isconnected to a substrate via one or more tethering structures, such as apedestal or array of pedestals. In an embodiment, for example, asubstrate is able to move relative to a tethered device or componentsupported by the substrate. In an embodiment, for example, a tethereddevice or component is able to move relative to a substrate supportingthe tethered device or component. In some embodiments, for example, atethered configuration of a device or device component decouplesmovement and/or deformation of the substrate from the device or devicecomponent. In some embodiments, for example, a tethered configuration ofa device or device component decouples forces generated by elongation,compression or deformation of the substrate from the device or devicecomponent. In some embodiments, less than 20%, optionally less than 5%,and optionally less than 1%, of the area of a bottom surface of a deviceor component is covalently bonded to the tethering structure connectedto the substrate.

“Nanostructured surface” and “microstructured surface” refer to devicesurfaces having nanometer-sized and micrometer-sized relief features,respectively. The relief features extend a length, x, from asubstantially contiguous plane of the device surface. Quantitativedescriptors of a structured surface include surface roughnessparameters, such as R_(max), R_(a), and normalized roughness(R_(a)/R_(max)), all of which may be measured by atomic force microscopy(AFM). R_(max) is the maximum height between a highest peak to a lowestvalley. R_(a) is the center-line-mean roughness, which is the average ofan absolute value of a deviation from a center line of a roughness curveto the roughness curve. The surface of a substrate or layer is“substantially smooth”, for the purposes of this disclosure, if thesurface has an R_(a) value of 100 nm or less. If the surface has anR_(a) value greater than 100 nm, the surface is considered to be a“structured surface” for purposes of this disclosure.

“Dielectric” refers to a non-conducting or insulating material. In anembodiment, an inorganic dielectric comprises a dielectric materialsubstantially free of carbon. Specific examples of inorganic dielectricmaterials include, but are not limited to, silicon nitride, silicondioxide and non-conjugated polymers.

“Polymer” refers to a macromolecule composed of repeating structuralunits connected by covalent chemical bonds or the polymerization productof one or more monomers, often characterized by a high molecular weight.The term polymer includes homopolymers, or polymers consistingessentially of a single repeating monomer subunit. The term polymer alsoincludes copolymers, or polymers consisting essentially of two or moremonomer subunits, such as random, block, alternating, segmented,grafted, tapered and other copolymers. Useful polymers include organicpolymers or inorganic polymers that may be in amorphous, semi-amorphous,crystalline or partially crystalline states. Crosslinked polymers havinglinked monomer chains are particularly useful for some applications.Polymers useable in the methods, devices and components include, but arenot limited to, plastics, elastomers, thermoplastic elastomers,elastoplastics, thermoplastics and acrylates. Exemplary polymersinclude, but are not limited to, acetal polymers, biodegradablepolymers, cellulosic polymers, fluoropolymers, nylons, polyacrylonitrilepolymers, polyamide-imide polymers, polyimides, polyarylates,polybenzimidazole, polybutylene, polycarbonate, polyesters,polyetherimide, polyethylene, polyethylene copolymers and modifiedpolyethylenes, polyketones, poly(methyl methacrylate),polymethylpentene, polyphenylene oxides and polyphenylene sulfides,polyphthalamide, polypropylene, polyurethanes, styrenic resins,sulfone-based resins, vinyl-based resins, rubber (including naturalrubber, styrene-butadiene, polybutadiene, neoprene, ethylene-propylene,butyl, nitrile, silicones), acrylic, nylon, polycarbonate, polyester,polyethylene, polypropylene, polystyrene, polyvinyl chloride, polyolefinor any combinations of these.

“Elastomeric stamp” and “elastomeric transfer device” are usedinterchangeably and refer to an elastomeric material having a surfacethat can receive as well as transfer a material. Exemplary elastomerictransfer devices include stamps, molds and masks. The transfer deviceaffects and/or facilitates material transfer from a donor material to areceiver material.

“Elastomer” refers to a polymeric material which can be stretched ordeformed and returned to its original shape without substantialpermanent deformation. Elastomers commonly undergo substantially elasticdeformations. Useful elastomers include those comprising polymers,copolymers, composite materials or mixtures of polymers and copolymers.Elastomeric layer refers to a layer comprising at least one elastomer.Elastomeric layers may also include dopants and other non-elastomericmaterials. Useful elastomers include, but are not limited to,thermoplastic elastomers, styrenic materials, olefinic materials,polyolefin, polyurethane thermoplastic elastomers, polyamides, syntheticrubbers, PDMS, polybutadiene, polyisobutylene,poly(styrene-butadiene-styrene), polyurethanes, polychloroprene andsilicones. In some embodiments, an elastomeric stamp comprises anelastomer. Exemplary elastomers include, but are not limited to siliconcontaining polymers such as polysiloxanes including poly(dimethylsiloxane) (i.e. PDMS and h-PDMS), poly(methyl siloxane), partiallyalkylated poly(methyl siloxane), poly(alkyl methyl siloxane) andpoly(phenyl methyl siloxane), silicon modified elastomers, thermoplasticelastomers, styrenic materials, olefinic materials, polyolefin,polyurethane thermoplastic elastomers, polyamides, synthetic rubbers,polyisobutylene, poly(styrene-butadiene-styrene), polyurethanes,polychloroprene and silicones. In an embodiment, a polymer is anelastomer.

“Conformable” refers to a device, material or substrate which has abending stiffness that is sufficiently low to allow the device, materialor substrate to adopt any desired contour profile, for example a contourprofile allowing for conformal contact with a surface having a patternof relief features.

“Conformal contact” refers to contact established between a device and areceiving surface. In one aspect, conformal contact involves amacroscopic adaptation of one or more surfaces (e.g., contact surfaces)of a device to the overall shape of a surface. In another aspect,conformal contact involves a microscopic adaptation of one or moresurfaces (e.g., contact surfaces) of a device to a surface resulting inan intimate contact substantially free of voids. In an embodiment,conformal contact involves adaptation of a contact surface(s) of thedevice to a receiving surface(s) such that intimate contact is achieved,for example, wherein less than 20% of the surface area of a contactsurface of the device does not physically contact the receiving surface,or optionally less than 10% of a contact surface of the device does notphysically contact the receiving surface, or optionally less than 5% ofa contact surface of the device does not physically contact thereceiving surface.

“Young's modulus” is a mechanical property of a material, device orlayer which refers to the ratio of stress to strain for a givensubstance. Young's modulus may be provided by the expression:

$\begin{matrix}{{E = {\frac{({stress})}{({strain})} = {\left( \frac{L_{0}}{\Delta\; L} \right)\left( \frac{F}{A} \right)}}},} & (I)\end{matrix}$where E is Young's modulus, L₀ is the equilibrium length, ΔL is thelength change under the applied stress, F is the force applied, and A isthe area over which the force is applied. Young's modulus may also beexpressed in terms of Lame constants via the equation:

$\begin{matrix}{{E = \frac{\mu\left( {{3\;\lambda} + {2\mu}} \right)}{\lambda + \mu}},} & ({II})\end{matrix}$where λ and μ are Lame constants. High Young's modulus (or “highmodulus”) and low Young's modulus (or “low modulus”) are relativedescriptors of the magnitude of Young's modulus in a given material,layer or device. In some embodiments, a high Young's modulus is largerthan a low Young's modulus, preferably about 10 times larger for someapplications, more preferably about 100 times larger for otherapplications, and even more preferably about 1000 times larger for yetother applications. In an embodiment, a low modulus layer has a Young'smodulus less than 100 MPa, optionally less than 10 MPa, and optionally aYoung's modulus selected from the range of 0.1 MPa to 50 MPa. In anembodiment, a high modulus layer has a Young's modulus greater than 100MPa, optionally greater than 10 GPa, and optionally a Young's modulusselected from the range of 1 GPa to 100 GPa.

“Inhomogeneous Young's modulus” refers to a material having a Young'smodulus that spatially varies (e.g., changes with surface location). Amaterial having an inhomogeneous Young's modulus may optionally bedescribed in terms of a “bulk” or “average” Young's modulus for theentire material.

Low modulus” refers to materials having a Young's modulus less than orequal to 1 MPa, less than or equal to 0.5 MPa, or less than or equal to200 KPa. A low modulus material may have a Young's modulus selected fromthe range of 1 MPa to 1 KPa, or 0.5 MPa to 1 KPa, or 200 KPa to 1 KPa,100 KPa to 1 KPa, or 50 KPa to 1 KPa.

“Bending stiffness” is a mechanical property of a material, device orlayer describing the resistance of the material, device or layer to anapplied bending moment. Generally, bending stiffness is defined as theproduct of the modulus and area moment of inertia of the material,device or layer. A material having an inhomogeneous bending stiffnessmay optionally be described in terms of a “bulk” or “average” bendingstiffness for the entire layer of material.

“Spatially offset” refers to an arrangement of features of a unit cellin a configuration where they do not completely overlap. For example, inone embodiment, features of a unit cell that are spatially offset unitcells are rotated with respect to one another. For example, inembodiments, features of a unit cell that are spatially offset unitcells are translated with respect to one another. For example, in oneembodiment, features of a unit cell that are spatially offset unit cellsare rotated and translated with respect to one another. In anembodiment, features of a unit cell that are spatially offset arepositioned in a plane or at a height different from one another. In anembodiment, features of a unit cell that are spatially offset possess adifferent inter-cell spacing length that that of other adjacent unitcells. In embodiments, spatially offset features of a unit cell arearranged such that the overall arrangement of all unit cells is not astraight line.

FIGS. 1A, 1B and 1C provide examples of two-dimensional spatialgeometries useful for circuits, devices and device components of theinvention. FIG. 1A illustrates three embodiments where electricalinterconnects or electrodes (fully bonded, selectively bonded, or freestanding) 101 are replaced by electrical interconnects or electrodeshaving spring-within-a-spring geometries 102 comprising arc-shapedserpentine segments. These embodiments illustrate the utility of aspectsof the invention for providing a geometry where multiple first-orderunit cells are arranged in a specific configuration to create asecondary geometry. The interconnects having the spring-with-a-springgeometry 102, in embodiments, adopt a space-filling or pseudo spacefilling configuration.

FIG. 1B further illustrates the concept of self-similar geometriesintegrated in one-dimensional serpentine horseshoe lines andtwo-dimensional serpentine horseshoe meshes. Curve 111 comprises aself-similar geometry, where the first order curved configuration,comprising a plurality of arc-shaped and spatially offset segments in aserial configuration arranged to form a second order curvedconfiguration, resulting in the formation of a spring-within-a-springgeometry. Curve 112 shows an embodiment where the arc angle of the firstorder configuration is increased, resulting in an increase in thespace-filling by the curve and an increase in overall path length andenhancement in stretchable mechanics, while still maintaining the secondorder curved configuration and spring-within-a-spring geometry. Curves113 and 114 illustrate overlapping arrangements of multiplespring-within-a-spring geometries, to create a stretchable 2D meshgeometry. The arc angle of the first order curvature of curve 114 isincreased as compared to curve 113, resulting in an overall greateramount of space filling and enhanced stretchability by curve 114. These1D lines and 2D meshes demonstrated in FIG. 1B are specific examples ofstretchable 1D and 2D constructs, and it will be apparent to one ofskill in the art that the geometries of the present invention aregeneral to a range of 1D and 2D structures.

FIG. 1C provides a schematic illustration of an iterative geometryshowing self-similarity at three different size scales for a Peano typecurve. In the top panel, the curve is characterized by a size dimensionL₁, corresponding to the side of a first order unit cell 121. The firstorder unit cell includes a folded geometry with arc-shaped segmentsarranged in a spatially offset configuration to provide an overallconnection between the lower left and the upper right of the unit cell,and including a pathlength longer than the straight line distancebetween lower left and upper right of the unit cell. In the middlepanel, the size of the unit cell is shrunk by about ⅓ to provide acharacteristic size dimension L₂, with 9 unit cells arranged in a serialconfiguration to provide a second order geometry 122 having a similarconfiguration to the geometry of the individual unit cell 121, providingan overall connection between the lower left and the upper right of thesecond order geometry and increased pathlength as compared to thepathlength of the first order unit cell and the point to point straightline distance from lower left to upper right. In the bottom panel, thesize of the unit cell is shrunk by about another ⅓ to provide acharacteristic size dimension L₃, with 9 second order unit cells (or 81first order unit cells) arranged in a serial configuration to provide athird order geometry 123 having a similar configuration to the secondorder geometry 122 and first order geometry 121, and again providing anoverall connection between lower left and upper right and significantlyincreased pathlength as compared to the pathlength of the first orderunit cell and the point to point straight line distance from lower leftto upper right. Here, the higher order geometries increase the spacefilling by the curve as compared with lower order geometries. For thisunit cell and curve configuration, additional iterations of thegeometries are contemplated. For example, as we iterate this particularfractal, we have the option of placing the individual N−1th subunitcells in either the x- or y-axis orientations, which leads to many Peanovariations. In an embodiment, for example, the characteristic sizedimension of the unit cell shrinking by about ⅓ for each successiveiteration and the number of first-order unit cells making up eachsuccessive order geometry increasing by a factor of 9, accompanied by anapproximately 9× increase in pathlength.

The invention may be further understood by the following non-limitingexamples.

Example 1: Fractal Design Concepts for Stretchable Electronics

Stretchable electronics provide a foundation for applications thatexceed the scope of conventional wafer and circuit board technologiesdue to their unique capacity to integrate with soft materials andcurvilinear surfaces. The ultimate scope of possibilities is predicatedon the development of device architectures that simultaneously offeradvanced electronic function and compliant mechanics. This example showsthat thin films of hard electronic materials patterned in deterministicfractal motifs and bonded to elastomers enable unusual mechanics withimportant implications in stretchable device design. In particular, thisexample demonstrates the utility of Peano, Greek cross, Vicsek, andother fractal constructs to yield space-filling structures of metals,polymers, and semiconductors, including monocrystalline silicon, forelectrophysiological sensors, precision monitors and actuators, andradio frequency antennas. These devices support conformal mounting onthe skin and have properties relevant to biomedicine, such asinvisibility under magnetic resonance imaging. The results suggest thatfractal-based layouts can be broadly developed as design strategies forhard-soft materials integration.

The field of stretchable electronics is of growing interest, motivatedboth by fundamental considerations in materials science and byapplication spaces in areas such as biomedicine. A core challenge is inachieving high performance electronic functionality with systems thatoffer low modulus, elastic responses to large strain deformations. Twoof the most successful approaches to this problem exploit advancedcomposites, in a generalized sense. The first involves dispersingconducting or semiconducting nanomaterials (i.e. nanowires, nanotubes orgraphene) into elastomeric matrices. Here, three-dimensional (3D)structures, including distributed networks of contacts, formspontaneously, but with limited direct control over key variables otherthan the overall loading fraction. The second exploits alternativeclasses of composites, created deterministically by using thin layers ofelectronic materials lithographically defined into two-dimensional (2D)filamentary mesh layouts. Advantages of this approach include theability to co-integrate multiple high-performance material platformswith high spatial resolution and engineering control, in ways that alsoallow proper electrical contacts, both internal and external to thesystem. An essential aim of both 3D random and 2D deterministiccomposites in stretchable electronics is to combine high loading of ahard component, for use in active devices, with overall mechanics thatis dominated by a soft matrix, for stretchable response. This goal isopposite to that associated with traditional composites engineering, andtherefore demands alternative approaches.

This example shows that concepts in fractal geometry, which are known todetermine behaviors in traditional 3D networks and which are pervasivein biological systems including the human body, can be successfullyexploited in 2D deterministic systems, with important functionalconsequences in advanced stretchable electronics. Fractal-basedstructures can be described by self-similarity: subdivision into smallsections yields pieces with geometries that resemble the whole. Comparedto previously explored networks of periodic serpentine shapes, fractaldesigns can be engineered to accommodate enhanced elastic strain along aselected dimension, and to support biaxial, radial, and otherdeformation modes. Additionally, the choices of topologies span a richrange, from lines to loops, capable of tailoring to specific electronicapplications through integration and interdigitation of multiplestructures. FIG. 2 presents six representative examples, from lines(Koch, Peano, Hilbert) to loops (Moore, Vicsek) and branch-like meshes(Greek cross). The results illustrate the diversity of possibilities,through both the finite element method (FEM) and experimentaldemonstration. The approximate fractal dimensions in thesefinite-iterative curves range from 1.5 to 2. The elastic tensile strainsachieved with these structures indicate they are suitable for use invarious stretchable devices, including the epidermal electronicplatform, with key advantages over previously described layouts.

The Peano curve provides a model system for examining the detailedmechanics of fractal-based motifs. Layouts using these or other designsfollow an iterative approach: to create the Nth Peano curve, nine copiesof the (N−1)th curve are linked together into a single line. Anillustration of the first three iterations of a Peano curve appears inFIG. 3 Panel (a). A physical interpretation that captures the underlyingmechanics follows by considering the first iterative curve geometry as asection of a one dimensional spring. Due to self-similarity, higherorder Peano curves contain spring-like motifs at multiple length scales(FIG. 7). Furthermore, each of these motifs can be oriented vertically(y-axis) or horizontally (x-axis). As such, the Peano geometry providesa design metric for systematically “folding” a straight line into acompact and stretchable layout. To further enhance the mechanics ofthese wires, arc sections replace the sharp corners in themathematically-defined Peano curves (FIG. 3 Panel (b)). Generally, theelastic mechanics of horseshoe serpentine structures improve withincreasing arc section angle (FIG. 19). Although the Peano curvesfeatured in FIG. 3 Panel (a) fill square areas, those of differentiterative orders can be linked together to fill spaces of arbitraryshape. As a demonstration, FIG. 3 Panels (c)-(f) show the word“ILLINOIS” formed with space filling wires, in which each letterconsists of a combination of first and second order Peano curves. Thefigure includes a detailed view of a section of the “N”. Electrodes canbe constructed in this manner to match, for example, particular featureson the body.

The Peano curves in FIG. 3 Panel (a) represent only one set ofvariations; there exist many others, because the (N−1)th unit cells ofany Nth order curve can be oriented either vertically or horizontally(FIG. 8). The second order curve has 272 unique layouts alone, each withdistinct mechanical properties. Numerical simulations of five differentsecond order layouts consisting of gold wires bonded on an elastomerprobe the relationship between mechanics and layout. The layouts containunit cells with orientations ranging from all vertical to all horizontal(FIG. 18). The maximum principal strain criterion defines the maximumelastic stretchability as the onset of plastic yield, consistent withestablished parameters of the constituent materials. FIG. 18 summarizesthis quantity, calculated for uniaxial deformation along the x- andy-axes. The results indicate that Peano layouts with unit cells alloriented in the same way maximize the uniaxial stretchability along theunit cell direction. The “half-and-half” Peano layout, which containsunit cells with alternating orientations, balances the maximum strainsupported along the x- and y-axis at 16% and 13%, respectively. Suchproperties are well suited for devices that stretch along both axes.Adjustments to the total unit cell size can enhance the mechanics ofthese structures (FIG. 20). The third order half-and-half layouts yieldstretchabilities along the x- and y-axis of 32% and 28%, respectively(FIG. 9). This improvement with iteration order is due to the combinedeffects of geometric scaling of the arc sections, increase in the lengthof the wire, and addition of higher order spring-like motifs. Thestretchability here is well above 20%, which is the general upper limitfor the elastic mechanics of skin.

Experimental structures consisting of second and third orderhalf-and-half Peano layouts of metallic wires that are clad withpolyimide and fully bonded to an elastomeric membrane compare well withthe numerical analysis (See Methods for details). The polyimide claddingretards strain localization in the metal, thereby enhancing sample yieldand consistency in mechanical testing. Three samples are tested for eachfractal dimension and axis of stretching. Mechanical characterizationinvolves measurements of resistance changes to a precision of ˜0.01 ohmswith a four point probe technique performed during uniaxial tensilestretching. In the elastic regime, the wire resistance before and aftercycling between states of no strain and increasing levels of maximumstrain (the difference defined as the “differential resistance”) doesnot change. At the onset of plastic deformation, the wire undergoes aresistance-dependent geometric change, leading to a small but measurabledifferential resistance. Traditional approaches to characterizingplasticity based on measurements of stress-strain response with atensometer are not suitable because the mechanics of the elastomericsubstrates in these systems dominate the response, by design.

FIG. 4a-4f and FIG. 9 summarize the results of these studies. Thedifferential resistances measured from individual representative devicesappear together in each plot. The error bars in the second and thirdorder Peano structures have magnitudes of 3.7e-5 and 3.3e-5,respectively; the Supplementary Section below discusses the sources ofthese errors. Simultaneous measurements of the local temperature accountfor and calibrate out changes in resistance due to backgroundtemperature fluctuations (see Supplementary Section below). Themeasurements show that for the second order Peano structures stretchedalong the x- and y-axis, the transition from elastic to plasticdeformation occurs in the range of 16-20% and 12-16%, respectively. Thethird order Peano structures undergo an elastic-to-plastic transition inthe range of 27-32% for stretching along both x- and y-axes. Thesevalues of uniaxial stretchability are consistent with the numericalanalysis. FEM strain maps (FIG. 4c and FIG. 4d ) identify the positionsof the mechanical “hot spots” in the samples, where strains areparticularly large and where failure occurs. Further enhancements indevice mechanics are possible by optimizing the detailed geometry ofthese hot spots.

Fractal-based structures bonded to pre-strained elastomers enable higherlevels of elastic deformation. A second order all-vertical Peanostructure fully bonded onto an elastomeric substrate with 40% pre-straindemonstrates the concept. The differential resistances for differentlevels of maximum applied strain appear in FIG. 10. Here, the transitionfrom elastic to plastic deformation occurs when the substrate isstrained in the range of 60-70%, which is significantly higher than thatin samples prepared without pre-strain. Schemes that use pre-strain canbe extended to biaxial, radial, or other types of deformation.

This concept of enhancing mechanics through the use of pre-strain isgeneral to a broad range of materials, including semiconductors. Thinfilms of single crystalline silicon nanomembranes (Si NM) with twodifferent second order Peano layouts and bonded onto 40% pre-strainedelastomeric substrates provide a proof of concept. In the pre-strainedstate, the calculated compressive stresses in the membrane are wellwithin the regime of elastic deformation for silicon. FIG. 4g and FIG.4h show microscale X-ray coherent tomography (microXCT) images of thesamples, along with corresponding FEM results. The findings indicatethat the all-vertical and half-and-half structures can be elasticallystrained by 105% and 63%, respectively, given a maximum principal strainof 1% in the silicon. The fracture points measured electrically fromhalf-and-half structures (FIG. 11) are consistent with the FEM results.Both the microXCT and FEM images reveal microscale buckling; this typeof deformation mode improves the ability of the Si NMs to dissipatestress. Such behavior persists only for a limited range of NMthicknesses. FEM simulations show that with increasing membranethickness, the NMs transition from a regime of wrinkling to microscalebuckling and finally to global buckling; furthermore, the microscalebuckling regime yields maximal elastic mechanics (FIG. 12a and FIG. 12b). As such, the optimization of the elastic properties of hard-softsystems requires careful attention to micro-mechanics.

These and other fractal layouts have utility for various applications.One is in skin-mounted electrodes for measuring electrophysiologicalprocesses in the brain (electroencephalograms (EEGs)), heart(electrocardiograms (ECGs)), and muscle (electromyograms (EMGs)). Tominimize impedance between the electrode and skin and to optimize themeasured signal-to-noise, electrodes require both conformal skin contactand high areal coverage. Electrodes that interface directly with neuronsadditionally benefit from having large perimeters within an area. Theseneeds can be addressed effectively using a variant of the Greek crossfractal, which consists of a hierarchy of cross structures that fillsspace in two dimensions (FIG. 14). This design embeds a high level ofconnectivity, which minimizes the resistance between any two points.Also, defects in the structure, such as line breaks, have a reducedprobability for significantly affecting device performance, which isdesirable for robust, long-term health monitoring in practice. Thesegeometries can be further designed to eliminate closed loops (FIG. 14),such that the edges of the electrode wire layout form a singlecontinuous line. A multifunctional device that incorporates a resistivetemperature sensing/heating element with an integrated electrode(recording, ground and reference components together) exploits thisfeature (FIG. 5a ). The temperature sensor consists of an insulated lineof metal running along the edges of the Greek cross electrode. FIGS.5b-5f show a representative device and its operation under differentmodes (heating, temperature sensing, ECG measurements). The impedancesand signal-to-noise measured with these dry electrodes compare favorablyto that of conventional gel-based electrodes, indicating that they aresuitable for high quality, skin-mounted, electrophysiologicalmeasurements. The precision of the temperature measurement (˜20 mK)compares well with that of an infrared camera.

Stretchable radio frequency antennas are another class of devices thatbenefit from concepts in fractal design. Fractal antennas have been atopic of interest because they can support multiband operation inspatial scales that are compact relative to the resonant wavelength.Appropriate choices of fractals offer not only this characteristic butalso attractive elastic mechanics when mounted on stretchablesubstrates. A Vicsek curve loop antenna, in which arc sections replacesharp bends, provides a model system. The antenna consists of coppertraces (3 μm thick) laminated with polyimide, and bonded onto a 1mm-thick elastomeric substrate. The copper thickness is comparable tothe skin depth of copper (˜2 μm) at gigahertz frequencies. The returnloss spectrum for the unstrained antenna displays a fundamental modenear 1.7 GHz (FIG. 6a ) with an impedance of 42 ohms at resonance. Thetotal length of the antenna at resonance is approximately λ₀/6, where λ₀is the free space wavelength, reflecting the compact nature of thisparticular fractal layout. As the device is strained, its fundamentalfrequency and input impedance slightly shift. Far-field measurements inan anechoic chamber provide additional information; data for thefundamental mode at 0% and 30% strain (FIG. 6c ) display a cleardipole-like pattern. The realized gain for both the unstretched andstretched devices ranges from −2 to 0 dB, which is slightly less thanthat for an ideal dipole due to ohmic losses in the thin copper wires.Simulations of the return losses and far field profiles are consistentwith the experiments (FIG. 17b ).

Another application that can benefit from the favorable RF propertiesand mechanics of fractal patterns is in electrode structures that arecompatible with magnetic resonance imaging (MRI). Copper foil samplesmounted onto a cylindrical phantom and scanned in a 3 Tesla Trio headscanner (Siemens Instruments) provide a demonstration. For purposes ofcomparison, the samples include three types of fractals, along with anunpatterned sheet, two variants of serpentine meshes, and superimposedvertical and horizontal lines. The fill fraction of the meshes and thefractal patterns are approximately the same (˜25%). For the magneticfield strength used here, the resonant RF pulse frequencies are ˜123MHz. FIG. 6d displays an axial cross-sectional scan obtained using spinecho (parameters are in the Supplementary Section below). Thewater-based phantom appears white due to its high hydrogenconcentration. The MRI image shows clear shadows in the vicinity of theunpatterned film and the mesh samples. Conversely, the fractal samplesexhibit no noticeable distortion or shadowing. Magnetostatic couplingbetween RF radiation and the samples, which yields circulating currentsand associated heat dissipation and signal loss, explain thesedifferences. The meshes couple to RF radiation because they consist ofhighly interconnected closed loops of metal; the fractals, on the otherhand, do not contain closed loops, do not couple to RF radiation, andare invisible in the MRI. This analysis suggests that fractal-baseddesigns provide routes to MRI-compatible skin-mounted or implantedelectronic devices.

In summary, fractal-based layouts create new design opportunities instretchable electronics, including a broad range of devices suitable forbiomedical systems. One of the challenges here is evaluating themechanical properties of these composite materials and rigorouslyidentifying their elastic and plastic mechanics. With the combination ofhigh precision electro-mechanical measurements and three-dimensional FEMsimulations, the fundamental mechanical responses and their dependenceon geometry can be understood and exploited for specific deformationmodes. This study suggests a general relationship between fractallayouts and mechanics that is broadly applicable to stretchablematerials engineering.

Methods.

Epidermal Device Fabrication. Fabrication of metal-based devices,including the temperature sensors and heaters, involves firstspin-coating and curing liquid polyimide (PI) on a UV-treatedPDMS-coated Si wafer. Electron beam evaporation yields layers ofchromium (4 nm) and gold (300 nm). Following metal patterning, etching,and the application of a second layer of polyimide, a photoresist maskand oxygen plasma etch define the polyimide-encapsulated device. All ofthe wires are 70 μm-wide and the polyimide layers are each 1.2 μm-thick.Water soluble tape (3M) retrieves the gold-polyimide structures, whichcovalently attach to an elastomer (0.5 mm-thick, Ecoflex or Solaris,Smooth-on Inc.) using an evaporated chromium-silica bonding layer. UVradiation pretreatment of the elastomer promotes covalent bondingbetween the silica and elastomer. An ACF cable (Elform, USA) attached tothe bonding pads of the device enable accessible electrical addressing.The same process, with an additional polyimide etch step, applies to theopen metal-based fractal electrodes for electrophysiological sensing.

An SOI wafer consisting of 260 nm-thick p-doped silicon on a 1 μm-thicksilicon dioxide layer is the starting point for the Si NM samples. Todetach the Si NMs from the handle wafer, an HF wet etch dissolves theunderlying silicon dioxide through an array of 3 μm-wide holes definedin the membranes. A PDMS stamp transfers the membranes onto a polyimidefilm, and a photoresist mask and dry etching process define themembranes into various Peano layouts. Gold wires electrically addressthe devices, and the same transfer steps described above finalize thedevices.

Antenna Fabrication. The starting material is copper foil (3 μm) on acopper carrier (35 μm; Olin Brass). Polyimide spun-cast and cured ontothe foil yield foil-polyimide laminates, which mount onto a PDMS-coatedsubstrate and enable copper carrier removal. A photoresist mask, wetcopper etch, and oxygen plasma dry etch pattern the antenna.

Fractal-based Metal Wire Simulations (Presented in FIGS. 2, 4 and 18).FEM yields the theoretical deformation, elastic-to-plastic transition,and fracture of the structures. Elastomeric substrates employ an 8-node,hexahedral brick solid element C3D8R in the FEM program, and the thinwires of PI/Au/PI-layered geometry employ a quadrilateral shell elementS4R with the section of composite layup. All of the wires are 70 μm-wideand consist of a 300 nm-thick gold layer sandwiched by 1.2 μm-thickpolyimide layers on each side. The total pattern areas are 7 by 7 mm andfully bond to a 0.5 mm-thick elastomer with a modulus of 50 kPa. Thesolid elements bond together physically and therefore share nodes withits adhered shell elements. An ideal elastic-plastic constitutiverelation with a Young's modulus of 78 GPa, Poisson's ratio of 0.44,yield strain of 0.3% and fracture strain of 5% describe the mechanicalbehavior of Au. The elastic-plastic transition is set when the maximumstrain of half the width of one section is beyond the yield strain of0.3%.

Electrode and Temperature Sensor Testing. The Greek cross electrodesrecord ECG signals from the torso. Scotch tape and an alcohol swabexfoliates the stratum corneum and removes dirt or particles to reduceeffects of hydration and surface impedance. Here, the ground electrode,located between the measurement and reference electrodes (˜7 mm apart atcenter-to-center distance), defines the common zero potential. Measuredsignals transmit wirelessly to a receiver, and commercial software usinga 60 Hz notch filter and low-pass Butterworth filters (BioRadio 150,Cleveland Medical Devices, USA) completes the analysis. The fractaltemperature sensors operate using the same four point probe techniquedescribed in the mechanical testing section. An IR camera and hot plateyields dV/dT used to calibrate the devices. The devices mount directlyonto the skin with no elastomeric backing layer with a spray bandagetechnique.

FIG. 18. Elastic mechanics of five different Peano-based wirestructures. Calculated stretchability of metal wires mounted on anelastomer in five different second order Peano layouts, given a maximumprincipal strain criterion of 0.3% in any section of the wires. Thelayouts range from “all-horizontal” (subunits are all oriented along thex-axis) to “all-vertical” (subunits are all oriented along the y-axis).The strain criterion defines the transition from elastic to plasticdeformation in the wires.

FIG. 19. Simulated uniaxial elastic stretchability for serpentine wiresas a function of arc solid angle. The inset of the middle column definesthe arc solid angle. The cross-sectional geometries and materials matchthose from FIG. 3, and all structures have R=620 μm and w=70 μm (definedin Figure S2). These simulations clearly display that elasticstretchability increases as a function of arc angle in these primitiveserpentine geometries. As such, deterministically defining the arcsection geometries in wire-type structures can help optimize themechanics.

FIG. 20. Simulated biaxial stretchability as a function of unit cellsize for half-and-half Peano structures. The cross-sectional geometriesand materials match those from FIGS. 4a-4h , and all structures havew=70 μm (defined in FIG. 8). The structures with unit cell sizes between1.5 mm and 4.5 mm display biaxial stretchabilities greater than 20% andare compatible with the elastic properties of skin.

Supplemental Information.

Analysis of Fractal Geometries with MicroXCT. Micro X-ray tomography(MicroXCT 400, Xradia) enables the imaging of the spatial topology ofthe various fractal structures. Experimental images in FIGS. 2 and 4demonstrate the structural details of fractal patterns from the MicroXCTsystem. Two magnifications, 0.5× and 4×, provide the whole views andmagnified views of the structures, respectively. Additional imagingparameters include a 40 KeV X-ray source and 200 μA current, with 600image frames recorded stepwise over 180 degrees. TXM Reconstructorsoftware (Xradia) reconstruct the images, and the visualization softwarepackage (Amira 5.4.2, Visage Imaging) yields the “gray-scale” images inFIGS. 2 and 4.

Mechanical Testing. The resistance of metal wires bonded to an elastomeris a function of both temperature and mechanical strain. Thecontribution of temperature is subtracted out to purely monitor changesin the metal resistance due to mechanical strain. The first step is tomeasure the resistance of the unstrained fractal samples as a functionof temperature to obtain dR/dT, using a hot plate and IR camera (FLIRSC7650). A linear fit using the least squares method with six resistancepoints in the range of 30° C. and 45° C. yields this calibration. Duringthe mechanical measurement itself, the temperature-adjusted resistance(R) follows from the continuously measured temperature (T) as:

$R = {R_{device} - {\frac{dR}{dT}\left( {T - T_{0}} \right)}}$R_(device) is the measured resistance of the device from the four pointmeasurement and T₀ is a constant reference temperature set before thestart of the experiment. FIG. 9 presents details of the experimentalsetup. FIG. 10 displays FEM images and the elastic-plastic transitionpoint for a pre-strained all-vertical Peano structure.

Temperature Sensor and Microheater Testing. Wires with Peano-basedlayouts have utility as the principal component in high-precisiontemperature sensors and heaters. Such sensors can be calibrated againstan infrared (IR) camera (A655SC, FLIR, USA), as performed above formechanical testing. The response of a third order half-and-half Peanosensor is plotted in FIG. 13. The noise of the analogue-digital (A/D)converter (V_(A/D))) and the electrical noise (V_(noise)) determine theprecision of the fractal temperature sensor using the expressions:

$T_{A/D} = {\left( \frac{dV}{dT} \right)^{- 1}V_{A/D}}$$T_{noise} = {\left( \frac{dV}{dT} \right)^{- 1}V_{noise}}$where dV/dT is from the temperature calibration. The precision of thefractal temperature sensor is 0.022° C. Mounting a device on a forearmand recording the temperature simultaneously with the sensor and an IRcamera illustrates applicability to measurements on the skin.

The same device can be used as a precision element for Joule heating.Infrared images of a device under 0% and 20% uniaxial strain show thatthe heating is uniform across the area of the device, which isindicative of the space-filling nature of the fractal construct (FIG.13c ). One application involves wound recovery, for which the deliveryof a controlled amount of heat to the wound vicinity leads to increasedlocal blood flow and vasodilatation and ultimately expedited recovery.The ability for these devices to function as both temperature sensorsand heaters enables the measurement of other quantities, such as thermalconductivity.

Fractal Antenna Design and Simulations. The box fractal antenna layoututilizes the two-dimensional box fractal illustrated in FIG. 13. Here,five versions of the (N−1)th geometry scale down and connect together toconstruct the Nth iterative geometry. A wire tracing around theperimeter of the box fractal creates the antenna layout; the sharpcorners are rounded to enhance the mechanics. The deformed antennaanalysis with HFSS simulations requires three steps: the undeformedantenna geometries import into Abacus, they numerically stretch with apredetermined strain, and this resulting geometry imports into HFSS forsimulation (FIG. 14). The frequencies and magnitudes of the calculatedS11 parameters, and the dipolar far field profiles, are consistent withthose measured experimentally.

MRI Imaging. The spin echo images use the following parameters:

TR 2000; TE 25; Averages 5; Slices 25; FOV 140×140 mm; Thickness: 3 mm;Flip angle: 60 deg; Resol: 256; Partial fourier: 5/8; Bandwidth: 130Hz/Px.

Example 2: Stretchable Batteries with Self-Similar SerpentineInterconnects and Integrated Wireless Recharging Systems

An important trend in electronics involves the development of materials,mechanical designs and manufacturing strategies that enable the use ofunconventional substrates, such as polymer films, metal foils, papersheets or rubber slabs. The last possibility is particularly challengingbecause the systems must accommodate not only bending but alsostretching, sometimes to high levels of strain (>100%). Although severalapproaches are available for the electronics, a persistent difficulty isin energy storage devices and power supplies that have similarmechanical properties, to allow their co-integration with theelectronics. This Example provides a set of materials and designconcepts for a rechargeable lithium ion battery technology that exploitsthin, low modulus, silicone elastomers as substrates, with a segmenteddesign of the active materials, and unusual ‘self-similar’ interconnectstructures. The result enables reversible levels of stretchability up to300%, while maintaining capacity densities of ˜1.1 mAh/cm². Stretchablewireless power transmission systems provide means to charge these typesof batteries, without direct physical contact.

Development of classes of electronic and optoelectronic technologiesthat offer elastic responses to large strain (>>1%) deformations hasaccelerated in recent years. Combinations of materials, device layouts,mechanics designs and manufacturing approaches are now beginning toemerge for realistic applications in areas ranging from wearablephotovoltaics to ‘epidermal’ health/wellness monitors, to sensitiverobotic skins, to soft surgical tools and electronic ‘eyeball’ imagingdevices. In many cases, stretchability represents a key, enablingcharacteristic. For many of these and other uses, a critical need liesin energy storage devices with similar physical properties, to allow fordirect and natural integration with the electronics. Many importantstorage devices have been developed with flexible characteristics,including supercapacitors and batteries. Here, sufficiently thingeometrical forms lead to flexibility, by virtue of bending inducedstrains (typically to values of ˜1% or less) that decrease linearly withthickness, for a given bend radius. Stretchability, on the other hand,represents a more challenging type of mechanics, in which the systemsmust accommodate large strain deformation (>>1%), typically of arbitraryform, including not only bending, but also twisting, stretching,compressing and others, and thickness is typically not a criticalfactor. Stretchable supercapacitors using buckled thin films of CNTs orCNT-soaked fibrous textiles, and stretchable non-rechargeable zinccarbon batteries based on conductive fabrics represent two examples.Although these technologies each have attractive features, none offersmultiple capabilities in recharging with high storage capacity,stretching to large levels of deformation (>100%), or establishingwireless electrical interfaces to external power supplies. The materialsand integration schemes provided in this example achieve thesecharacteristics in a type of lithium ion battery that exploits segmentedlayouts and deformable electrical interconnects in specialized,‘self-similar’ geometries. The resulting devices offer biaxialstretchability up to strains of 300%, with capacity densities of ˜1.1mAh/cm², and little loss in capacity for up to 20 cycles of recharging.The designs also allow integration of stretchable, inductive coils toenable charging through external supplies without the need for physicalconnections. This set of attributes satisfies requirements for manyapplications that are being contemplated for stretchable electronics.

Results.

Battery design. The devices of this example exploit pouch cells in whicharrays of small-scale storage components are connected by conductingframeworks with extraordinary stretchable characteristics. A schematicillustration of the system, an exploded view of the multilayerconstruction of a unit cell, and a representation of the ‘self-similar’interconnect geometries appear in FIGS. 21A, 21B, 21C, and FIG. 25below. The current collectors consist of photolithographically patternedcircular disks of aluminum (600 nm) and copper (600 nm). Layers ofpolyimide (PI; 1.2 μm) encapsulate interconnecting traces between thesedisks in a way that places the metals close to the neutral mechanicalplane (FIG. 21D and FIG. 21E, left panel). Thin (0.25 mm), low modulus(60 KPa) sheets of silicone elastomer form top and bottom substratesthat support these structures (FIG. 21D and FIG. 21E, middle panel) andother components of the batteries. The overall construct consists of asquare array of 100 electrode disks, electrically connected in parallel.Molded pads of slurries based on LiCoO₂ and Li₄Ti₅O₁₂ serve as activematerials at the cathode and anode, respectively (FIG. 21E and FIG. 21E,right panel, and FIG. 26). The two sheets laminate together in a waythat involves spatial offsets between the active materials to avoidelectrical shortage between them and to eliminate the need for aseparator. A spacer, made of the same silicone elastomer and appliedaround the periphery of the system, prevents direct contact of the topand bottom sheets. A gel electrolyte injected into the gap providesmedia for ionic transport. Thin encapsulating layers of an acryloxyperfluoropolyether elastomer bonded to the outer surfaces help toprevent moisture from diffusing into the battery and solvents in the gelfrom leaking out. Long term operation requires more sophisticatedpackages consisting, for example, of buckled bilayer sheets ofaluminum/polyimide that bond to the outer surfaces of the battery (FIG.27). The materials and fabrication details appear in the Methodssection.

The devices must accommodate two competing design goals: (1) achievinghigh areal capacity, which requires large coverage of the activeregions, and (2) affording high mechanical stretchability, whichrequires large distances between these regions. Strategic features ofrelief on the elastomer substrates provide a partial solution to thischallenge, as demonstrated recently in photovoltaic modules. Adisadvantage is that levels of stretchability beyond ˜30% can bedifficult to achieve without sacrificing coverage. Here, a different,but complementary, approach is taken in which the focus is on deformableinterconnects with advanced designs. In particular, layouts areintroduced that use ‘self-similar’ structures of wires in serpentineconfigurations to offer, simultaneously, high system-levelstretchability, and low interconnect resistances. A conventionalserpentine consists of circular arcs connected by straight lines.‘Self-similar’ designs follow from iteratively applying this basicgeometry, beginning with a unit cell as illustrated schematically in thered box of FIG. 21C. Here, reducing the scale of the cell, and thenconnecting multiple copies of it in a fashion that reproduces the layoutof the original cell geometry corresponds to one iteration. The yellowline in FIG. 21C represents a 2nd order serpentine geometry, created inthis fashion. Although higher orders can be designed and implementedeasily, the 2nd order construct satisfies requirements for theapplications considered here, as described in the detailed experimentaland theoretical study below.

Mechanical Characteristics of the ‘Self-Similar’ Interconnects.

Three-dimensional (3D) finite element analysis (FEA) (details below) andexperimental measurements illustrate the essential mechanics. Testsamples fabricated for this purpose consist of free-standing, multilayertraces, with materials and multilayer stack designs (PI (1.2 μm)/Cu (0.6μm)/PI (1.2 μm)) that match those used in the batteries, betweencircular pads that bond to posts molded onto underlying elastomersubstrates. The self-similar geometry leads to hierarchical bucklingphysics that ensure ultra-low strains in the materials, even underextreme stretching. For the entire range of tensile strains examined,from 0% to 300%, the configurations predicted by FEA agree remarkablywell with optical images collected during the experiments, as shown inFIG. 22. Both symmetric and anti-symmetric buckling modes exist (seeFIGS. 28A and 28B for detailed illustrations of the two modes). Thetrace consists of three columns of serpentine wires connected by twohorizontal straight lines. We refer to the construct that corresponds tothe ‘short’ wavelength serpentine within each column as the 1st level;the 2nd level corresponds to the large-scale serpentine shape, with‘long’ wavelength. For the symmetric buckling mode (FIG. 28A), the leftand right columns undergo mainly an overall bending deformation alongthe vertical direction, resulting in the collective upward motion of theentire middle column of serpentine wires. In this sense, theout-of-plane displacement is symmetric with respect to the center line(x=0) in the “Front view” of FIG. 28A. For the anti-symmetric bucklingmode (FIG. 28B), the serpentines in the left and right columns mainlyundergo an overall twisting deformation along the vertical direction.Here, the two ends of the middle serpentine move in opposite directions(i.e. one moves up, and the other moves down). In this case, theout-of-plane displacement is anti-symmetric with respect to the centerline (x=0) in the “Front view” of FIG. 28B. The critical bucklingstrains obtained by FEA for the symmetric (0.078%) and anti-symmetric(0.087%) modes are much lower than those (>0.172%) for all otherbuckling modes. This result is consistent with experimental observationof only these two modes. In both cases, the physics associated withstretching involves a mechanism of “ordered unraveling”, which begins atthe 2nd level, at a well-defined, critical buckling strain, ˜0.08% forthe example investigated here. Next, the 2nd level gradually “unravels”via bending and twisting as the applied strain increases from 0.08% to˜150%, during which there is essentially no further deformation in the1st level. The motions in the 1st level start when the 2nd level isalmost fully extended, corresponding to an applied strain of ˜150% inthis case. As the “unraveling” of the 1st level serpentine approachesits end, the strain in the materials begins to increase rapidly, therebydefining the practical limit in stretchability.

For applied strains below this limit, the deformation mechanisms ofordered unraveling processes ensure low levels of strain in thematerials (FIG. 29). For a representative failure strain of 1% forcopper, FEA predicts a stretchability of 321%, which is in goodagreement with the experimental observations(300%<ε_(stretchability)<350%). (Simulations suggest that the copperreaches its failure point before the PI.) For reversible behavior (i.e.,the interconnects return to their initial configuration after release),the maximum material strain must be less than the yield strain. For arepresentative yield strain of 0.3% for copper, FEA suggestsreversibility for applied strains up to ˜168%. This value is lower thanexperimental observations, where reversibility occurs even for strainsof between 200% and 250% (FIG. 22). The likely explanation for thisdiscrepancy is that yield occurs first in only small portions of theinterconnect (e.g., one element in the FEA). In this case, the effectson reversibility might not be easily observed in experiments.

These levels of stretchability (>300%) and reversibility (>200%)significantly exceed those of previous reports in stretchable batteriesand/or battery electrodes; they are also greater than those of any otherreports of stretchable interconnects that use lithographically definedpatterns of conventional metals. The importance of the self-similardesigns can be assessed through comparisons of stretchability tootherwise similar, but conventional serpentine structures: the formerexhibits a stretching range of 321%, while the latter is 134%,determined by FEA (FIG. 30). Furthermore, even for the same total length(l_(total)), span (L), amplitude (h), and cross section (width wandthickness t), the self-similar design again outperforms the conventionalserpentine, both in stretchability (809% vs. 682%) and reversibility(528% vs. 284%) (FIG. 31). We note that in all cases of uniaxialstretching, the Poisson effect leads to compression in the orthogonaldirection. The buckling profiles in these regions have behaviors thatare consistent with FEA (FIG. 32).

Electrochemical and Mechanical Behavior of the Battery.

After choosing a set of dimensions that offers excellent system levelstretchability, with good areal capacity density, and modestinterconnect resistance, the best electrical performance was observedfor layouts in which the diameters of the disks for the cathode andanode are 2.20 mm and 1.58 mm, respectively, and the offset distancesare 0.51 mm. This configuration corresponds to an areal coverage of 33%for the cathode, 17% for the anode, and 50% for the entire battery (inthe undeformed configuration) (FIGS. 33A and 33B). The interconnectshave thicknesses of 600 nm and widths of 50 μm. For these parameters,the resistance between adjacent disks is 24Ω, and that between theconnection lead and the most distant disk is 45Ω. The leads for externalconnection are thin and narrow to avoid strain at the interface, andfacilitate connection to flexible (but not stretchable) cables thatconnect to external characterization equipment. The gel electrolytecombines the flow properties of viscous liquids with the cohesiveproperties of a solid, thereby allowing it to accommodate large strainswhile maintaining ionic conduction pathways.

Electrochemical properties of the battery electrodes without and with300% uniaxial strain appear in FIG. 23A. The results show two welldefined plateaus at around 2.35 V corresponding to potentials of Co³⁺/⁴⁺and Ti⁴⁺/³⁺ redox couples. The thickness of the LiCoO₂ (specificcapacity 145 mAh/g) at each unit is ˜120 μm, corresponding to a mass of˜95 mg, and thus areal capacity density of 1.1 mAh/cm² at acharge/discharge rate of C/2. The mass of Li₄Ti₅O₁₂ (specific capacity160 mAh/g) is ˜90 mg, which corresponds to 5%-10% more anode capacitythan cathode. Slurry disks with thicknesses larger than those describedhere yield improved areal capacity density, but with reduced ratecapability due to the concentration polarization in the disks. Theoutput resistance of the battery is ˜70Ω (FIG. 34), and the leakagecurrent is 1-10 μA. The leakage current arises from three main sources:(i) the reverse biased Schottky diode, (ii) internal ohmicself-discharge between the slurry disks at the anode and cathode and(iii) Faradaic effects, including shuttle reactions associated withimpurities in the slurry materials, residual oxygen and/or moisture.Experimental results presented below and in FIG. 35 show that use ofseparators and enhanced packaging schemes can reduce the capacity lossfrom 161 μA·h to 23 μA·h in 44 hours. FIG. 23B shows the coulombicefficiency (red) and cycling performance (black) of the encapsulatedbattery. The coulombic efficiency rises from ˜60% for the first cycle toover 90% after three cycles. The initial loss can be attributed to theformation cycle, during which a solid-electrolyte-interphase forms, andlithium is consumed in side reactions with impurities in theelectrolyte. The gradually degrading capacity retention results ratherfrom the cycle fade (FIG. 36) but more likely from the calendar fade dueto some combination of reaction with residual water in the packagingmaterials, moisture penetration, and electrical discontinuity of slurryparticles that detach from the disks (which are not hot-pressed), andcan be sometimes observed in the electrolyte gel. Varying the depth ofdischarge from 100% to 75% did not have a significant effect on thedegradation characteristics (FIG. 37). Further increasing the bakingtemperature and optimizing the composition of the slurries, such asincreasing the binder ratio, could reduce the latter behaviors. Improvedconditions for device assembly could reduce effects of the former. FIG.23C shows the output power of the battery, when connected to a resistor(2020Ω), during biaxial stretching and releasing. The slight decrease inoutput power with strain likely results from increased internalresistances that arise from the significantly increased separationsbetween slurry disks with strains at these large levels. The batteryprovides sufficient power to operate commercial light emitting diodes(LEDs), with turn on voltages of 1.7 V (FIG. 38), as shown in FIG. 23D.The battery could be stretched for up to 300% (FIG. 23E), folded (FIG.23F), twisted (FIG. 23G), and compliant when mounted on human skin (FIG.23H) without noticeable dimming of the LED. Furthermore, FEAdemonstrates that the effective modulus (66.8 KPa) of the full compositestructure of the battery is only slightly higher than the modulus (60.0KPa) of substrate materials (Ecoflex). As a result, the battery is notonly stretchable but also exceptionally soft and compliant. The modulusis, in fact, lower than that of the human epidermis (140-600 KPa),thereby offering the potential for integration onto the skin andbiological tissues, without significant mechanical loading.

Stretchable Wireless Charging System for the Battery.

In many practical cases such as embedded devices, the ability to chargethe battery without establishing physical connections to externalsupplies can be valuable. Even in systems where the charging terminalsare accessible, such as in skin-mounted devices, there is value inwireless charging, simply because the process of establishing physicalcontacts can be mechanically destructive to thin, stretchable devices(or to the underlying soft tissue). Approaches that involve physicalcontact also have the danger of electrical shock to surroundingmaterials (e.g. the skin itself). The versatility of the materials anddesigns enable integration of wireless power transmission systems,monolithically with the battery itself. The design and an actual deviceappear in FIGS. 24A, 24B, respectively. A secondary coil couples theelectromagnetic flux from a primary coil, and a Schottky diode providesrectification. The Schottky diode (packaged in epoxy, with a modulus of˜4.0 GPa) has a modulus of more than 4 orders of magnitude larger thanthat of the substrate (made of Ecoflex, modulus ˜60 KPa), but its size(length 0.62 mm, width 0.32 mm, and height 0.31 mm) is only a fewpercent (˜2%) of the overall size (˜30 mm×˜20 mm) of the wirelesssystem. As a result, the influence on the overall stretchability isstill negligible, as demonstrated by finite element simulations shown inFIGS. 39, 40A and 40B. The capacitor smooths oscillations in the outputvoltages; its small size and thickness enable natural integration intothe overall system. Larger capacitors can smooth the oscillations to aneven greater extent (FIG. 41). The coil and rectifier add a seriesresistance of 2.3 KΩ2 (FIG. 42), which functions as a parallelresistance with the secondary coil, shunting away current from thebattery. The resistance of the serpentine secondary coil is 1.92 kΩ/m; acoil with similar overall geometry but without the serpentine shape iscalculated to be 1.22 kΩ/m. Improving the efficiency of the chargingsystem can be achieved by increasing the width and thickness of thewires, but at the expense of reduced stretchability and increasedmodulus. Specific application requirements will define the righttradeoffs. In this case, the output power from the primary coil was 187mW. With a working distance of 1 mm between the primary and secondarycoil, the power received on the secondary coil is 9.2 mW, correspondingto an efficiency of 4.9%. The power coupling efficiency of the wirelesscharging system depends strongly on the resistance of the serpentinereceiver coil. Increasing the thickness to 7 μm and using copperimproves the efficiency from 4.9% to 17.2%. At this thickness, the coilretains stretchability to strains of 25%. Data and images are describedbelow. The capacitor has a capacitance of 1.7 nF, in a structure thatuses a 1.2 μm thick layer of polyimide as the dielectric, with a layerof thiol molecules on the bottom Au electrodes to enhance adhesion. FIG.24C shows the input and output of this wireless power transmissiondevice. An input voltage at a frequency of 44.5 MHz matches theself-resonant frequency of the secondary coil, which is dependent on thecoil area, number of turns, distance between each turn, and wireresistance. For a peak-to-peak input voltage of 9.1 V (FIG. 24C blackcurve), the DC output voltage is 3.0 V (FIG. 24C red curve). Thecharging curves of a small scale battery using the wireless coil appearin FIG. 24D. The battery voltage (FIG. 24D orange curve) rises to 2.5 Vin about 6 mins. The charging current in the circuit (FIG. 24D bluecurve) decreases from 0.5 mA to below 0.2 mA. We used a partialdifferential equation to model the charging circuit, and a numericalprogram to calculate the charging current curve. Simulation of thisprocess agrees well with the experimental data (see below and FIG. 40).

Discussion.

The materials and schemes described in this example provide routes toenergy storage devices and wireless charging systems with forms andproperties that are attractive for powering stretchable electronic andoptoelectronic devices. The slurry materials themselves are deployed inways (a soft lithographic type casting process) that allow naturalintegration with unusual materials (low modulus silicone rubber sheets,embossed with surface relief). The stretchable interconnects exploit a‘self-similar’ design that offers unique, ‘spring within a spring’mechanics. The consequence is a level of stretchability that is morethan 4× larger than previous reports, even while, at the same time,enabling coverages of active materials as high as 50%. The combinationof these two aspects, together with comprehensive and experimentallyvalidated models of the underlying mechanics, leads to a technology,i.e. a stretchable, rechargeable battery, which offers much differentcharacteristics than anything that has been previously possible. As anadditional advance, we introduce integrated stretchable, wirelesscharging systems that offer physical properties similar to those of thebatteries.

The slurry chemistries, separator materials, and stretchable,air-impermeable packaging materials can be selected to provide highdevice performance. The self-similar serpentine interconnect structurepossesses a geometry of space filling curve, and a buckling physics ofordered unraveling. This type of interconnect structure has immediate,and general utility, suitable for any class of stretchable technologythat combines hard and soft materials. The topology-level interconnectgeometry simultaneously provides for large mechanical stretchability andlow electrical resistance. Wireless power transfer efficiency can beimproved by reducing the coil input resistance, maximizing the mutualinductance between the primary and secondary coils, and increasing theself-resonant frequency of the coils. Regulation circuits may beincorporated to avoid over-charging the battery.

Methods.

Fabrication of electrodes and mechanical testing of self-similarinterconnects: Sequential spin casting defined a bilayer of poly(methylmethacrylate) (PMMA 495 A2, 3000 rpm for 30 s, baked on a hotplate at180° C. for 2 mins) and then a layer of polyimide (PI, frompoly(pyromellitic dianhydride-co-4,4′-oxydianiline) amic acid solution;4000 rpm for 30 s, baked on a hotplate at 150° C. for 4 mins and then ina vacuum oven at 10 mT and 250° C. for 1 h) on a silicon wafer. Thecathode and anodes consisted of 600 nm thick layers of Al or Cu,respectively, deposited by electron beam evaporation onto the PI.Photolithography (AZ5214E) and etching (Type A aluminum etchant onhotplate at 50° C. for 2 min; Type CE-100 copper etchant at roomtemperature for 10 s; Transene Company) defined patterns in thesemetals. After removing the residual photoresist, spin coating formed anadditional layer of PI over the entire structure. Next, photolithography(AZ 4620, 2000 rpm for 30 s, baked at 110° C. for 4 mins) and oxygenplasma etching (300 mT, 20 sccm O₂, 200 W for 10 mins) patterned thelayers of PI in a geometry matched to the metal traces.

Immersion in hot acetone partially removed the underlying PMMA layer,thereby allowing the entire structure to be retrieved from the siliconwafer onto the surface of a piece of water soluble tape (3M, Inc.).Electron beam evaporation of Ti (5 nm)/SiO₂ (50 nm) through a shadowmask formed backside coatings aligned to the metal disks³³. Thin (250μm) silicone substrates (Ecoflex, Smooth-On) were prepared by mixing thetwo components in a 1:1 weight ratio, spin-casting (300 rpm for 30 s)the resulting material into a petri dish and then partially curing it(30 mins at room temperature). Next, spin-casting (3000 rpm for 30 s) anallyl amide functional perfluorinated ether (DuPont), and then curing itunder ultraviolet (UV) light for 30 mins formed a top encapsulationlayer. The other side of the Ecoflex substrate was surface-activatedunder UV light for 5 mins. Laminating the electrode structures onto thissurface led to strong bonding, upon contact. The water soluble tape wasremoved by immersing the substrate in tap water for overnight. As afinal step, the electrodes were dipped in 1 mM HCl to remove oxides fromthe surfaces of the metals.

Mechanical testing of the self-similar interconnects was performed witha customized uniaxial stretcher. To ensure that the interconnects weredecoupled from the substrate, each disk was mounted on top of a post(250 mm in height) molded into the silicone substrate. Images and videoof the deformed interconnects were collected with a digital single-lensreflex camera.

Patterned Moulding of Slurries and their Integration with CurrentCollecting Electrodes:

Photolithography (AZ 4620, 7-8 μm thick) and inductively coupled plasmareactive ion etching (ICP RIE) defined arrays of cylindrical wells onthe surface of a silicon wafer. The conditions were chosen to yieldsloped sidewalls, which are important for effective transfer of theslurries, as described subsequently. Washing with acetone removed thephotoresist. A layer of polytetrafluoroethylene (˜200 nm) conformallydeposited using the ICP RIE tool served as a coating to preventadhesion. The slurry materials consisted of lithium cobalt oxide orlithium titanium oxide, acetylene black, and polyvinylidene fluoride,mixed in a weight ratio of 8:1:1 in a solvent of N-methyl-2-pyrrolidone(NMP) for the cathode and anode, respectively. The mixture was stirredfor overnight, and then mechanically scraped across the etched surfaceof the silicon wafer. The cylindrical wells, filled with slurry in thismanner, were baked on a hotplate at 90° C. for overnight, and thenretrieved onto the surface of a piece of water soluble tape. The bakingconditions were selected carefully to enable retrieval of the slurrywith high yield. Registering the tape to the metal electrodes ensuredthat the molded slurry aligned to the metal disks. Immersion in tapwater for overnight dissolved the tape. Baking the substrates at 170° C.for overnight in a vacuum oven removed the moisture and improved thestrength of bonding between the slurry and the metal.

Assembly and Electrochemical Testing of the Battery:

Anisotropic conductive films, hot pressed onto the metal electrodes,served as points for external electrical connection. Application ofSylgard Prime Coat (Dow Corning, Inc.) to the periphery of thesubstrates prepared them for bonding. A thin silicone spacer layer (500μm thick) at the periphery prevented direct contact as the two sheetswere laminated together. A lateral spatial offset prevented electricalshorting between the active regions. The edges were sealed with anadditional application of Ecoflex followed by baking on a hotplate (90°C. for 2 h). The gel electrolyte consisted of a mixture of 100 g lithiumperchlorate, 500 ml ethylene carbonate, 500 ml dimethylcarbonate, and 10g polyethylene oxide (4×106 g/mol), prepared in an argon filled gloveboxas a homogenous gel. This material was injected into the battery using asyringe through an edge.

A BioLogic VMP3 electrochemical station with a cutoff voltage of 2.5-1.6V at room temperature was used to charge and discharge the as-fabricatedand stretched battery electrodes, and to evaluate cycling behavior ofthe full, integrated battery. Areal capacity density was calculatedbased on the active region. The output power was monitored with thebattery connected to a 2020Ω resistor, using an ammeter. Values ofcurrent were recorded as a function of strain applied to the battery.

Fabrication and Testing of the Wireless Coil:

A silicon wafer, coated with layers of PMMA and PI using steps describedpreviously, served as a substrate for deposition of Cr (5 nm)/Au (500nm) by electron beam evaporation. Photolithography (AZ 5214E) andetching (Transene Company) defined patterns in the geometry of thebottom electrode of the capacitor and associated contact lines. Removingthe photoresist with acetone, and then immersing the substrate in a 1 mMpoly(ethylene glycol) methyl ether thiol solution in isopropanol for 15mins served to enhance the adhesion and coverage of a second layer of PIspin-cast (4000 rpm 30 s) and cured (on hotplate at 150° C. for 4 minsand then in vacuum oven at 250° C. for 1 h) on the electrodes. Thislayer of PI served as the dielectric for the capacitor. Photolithography(AZ 4620, 2000 rpm for 30 s, baked at 110° C. for 4 mins) defined a maskfor etching vias through the PI layer, as points of connection betweenthe coil and the bottom electrode of the capacitor. After immersion inacetone to remove the photoresist, sputter deposition formed a conformallayer of Cu (600 nm) over the entire surface, including the sidewalls.Photolithography (AZ 5214E) and etching defined the coil and the othertop electrode of the capacitor. After removing the resist, a thirdspin-cast layer of PI formed a coating on the Cu electrodes. An oxygenplasma etching through the three PI layers in a patterned geometrydefined the final device layout. Release by immersion in hot acetonepartially removed the underlying PMMA, to enable the release of theentire structure onto water soluble tape. Deposition of Ti (5 nm)/SiO₂(50 nm) and lamination onto the UV activated surface of an ecoflexsubstrate led to strong bonding. After the water soluble tape wasremoved, a Schottky diode chip (Digikey BAT 62-02LS E6327) was bondedbetween the coil and the capacitor using silver epoxy. The forward inputresistance is ˜500Ω, and the rectification ratio is ˜1×104 at a biasvoltage of ±1 V.

High frequency alternating current signals were generated by a KEITHLEY3390 50 MHz arbitrary waveform generator. The input and outputcharacterization of the wireless coil were performed using an Agilentinfiniium DSO8104A oscilloscope (1 GHz, 4 channels). The wirelesscharging voltage and current to the battery were monitored using aBioLogic VMP3 electrochemical station.

Supplementary Information.

Fabrication of a stretchable encapsulating layer, consisting of abuckled sheet of Al/PI on a silicone substrate. The first step involvedfabrication of a trilayer of PMMA/PI/Al on a silicon substrate, usingprocedures similar to those employed for the Al battery electrodes.Photolithography with AZ5214E and wet etching the Al defined the lateraldimensions of the PI/Al sheet. Next, oxygen plasma etching (300 mT, 20sccm O₂, 200 W for 5 mins) removed the PI layer in the exposed regions.Immersion in hot acetone partially removed the underlying PMMA layer,thereby allowing the entire structure to be retrieved from the siliconwafer onto the surface of a piece of water soluble tape (3M, Inc.).Electron beam evaporation of Ti (5 nm)/SiO₂ (50 nm) formed backsidecoatings. On a separate substrate, 500 μm thick silicone sheets(Ecoflex, Smooth-On) were prepared, then removed and prestrainedbiaxially to a level of ˜30% and then fixed by lamination onto a glassslide. The silicone surface was activated by exposure to UV-inducedozone for 5 mins. Laminating the PI/Al bilayer onto this surface led tostrong bonding, upon contact. The water soluble tape was removed byimmersing the substrate in tap water for overnight. Peeling the entireassembly away from the glass slide released the prestrain, and led to anexpected pattern of buckling. In this configuration, the overall systemcan be stretched to strains as large as those equal to the prestrain.

Mechanical Analyses of “Island-Bridge” Self-Similar ElectrodeStructures:

FEA. Full three-dimensional (3D) FEA is adopted to analyze thepostbuckling behaviors of “island-bridge” self-similar electrodestructures under stretching and compression. Due to the structuralperiodicity of the electrode, a representative unit cell was studied,and its detailed geometry is shown in FIG. 25. The circular island ofthe electrode is mounted on a post (height 250 μm) molded on the surfaceof a silicone substrate (ecoflex; thickness 500 μm). The metalinterconnect (thickness 0.6 μm), is encased, top and bottom, by a thinlayer of polyimide (PI, thickness 1.2 μm for each layer). The elasticmodulus (E) and Poisson's ratio (v) are E_(ecoflex)=0.06 MPa andv_(ecoflex)=0.49 for ecoflex; E_(Cu)=119 GPa and v_(Cu)=0.34 for copper;E_(Al)=70 GPa and v_(Al)=0.35 for aluminum; and E_(PI)=2.5 GPa andv_(PI)=0.34 for PI. Eight-node 3D solid elements and four-node shellelements were used for the ecoflex and self-similar electrode,respectively, and refined meshes were adopted to ensure the accuracy.The evolution of deformed configurations with applied strains areobtained from FEA for both stretching and compression, as shown in FIG.22 and FIG. 32, respectively. Good agreement between FEA and experimentresults can be found. Here, we take the case of copper as ademonstration of the buckling profiles. The results for the aluminiumlayer are similar. For the comparison of stretchability andreversibility between self-similar and serpentine interconnects (FIG.31), the key geometrical parameters are kept the same, including thetotal length (I_(total)=16.77 mm), span (L=1.6 mm), amplitude (h=0.4mm), width (w=30 μm), and thickness (t=3.0 μm). The aluminuminterconnect (thickness 0.6 μm) is encased by two thin layers ofpolyimide (thickness 1.2 μm for each layer). FIG. 31 demonstrates thatover the entire range of stretching from 0% to ˜800%, the strain levelof the self-similar interconnect is always lower than the serpentineone. The stretchability (809%) and reversibility (528%) of theself-similar design, are higher than those (ε_(stretchability)=682%,ε_(reversibility)=284%) of the simple serpentine design.

Battery Leakage Current Analysis.

The leakage current arises from three sources. The first source iscurrent through the reverse biased Schottky diode. This current is ˜0.2μA, and is relatively constant throughout the lifetime of the battery.Schottky diodes with smaller reverse current are available; such devicescan reduce this contribution to the leakage.

The second source is the internal ohmic self-discharge current betweenthe slurry disks at the anode and cathode. This contribution arises fromfinite electronic conductivity of the electrolyte and any parasiticphysical contacts between the slurry disks at the cathode and anode.These losses can be dramatically reduced by electrolyte materials withenhanced purity and implementing separators. New experiments reveal thelatter effects quantitatively. FIG. 35 shows the voltage decay andleakage current curves for otherwise similar batteries with and withouta commercial separator (Celgard). This component reduces the capacityloss from 161 μA·h to 88 μA·h in 44 hours.

The third source is from current produced by Faradaic effects, includingshuttle reactions associated with impurities in the slurry materials,residual oxygen and/or moisture. Improving the air-impermeability of thepackaging materials can reduce currents that arise from such reactions.New experiments show that sealing the battery in an Al pouch (which canbe configured in a wrinkled configuration to allow stretching) reducesthe capacity loss from 161 μA·h to 62 μA·h. Combining the separator andthe Al pouch packaging, suppresses the capacity loss to 23 μA·h. FIG. 35summarizes all of these findings.

Discrete Schottky Diode Stretching Behavior Analysis.

From a practical standpoint, we observe no significant constraints onthe overall mechanical properties of the integrated system, due to thecomparatively small size of the diode. In particular, although theSchottky diode, which is encapsulated in epoxy which has a modulus of˜4.0 GPa, is effectively more than 4 orders of magnitude larger thanthat of substrate (made of Ecoflex, with a modulus of ˜60 KPa), itsdimensions (length 0.62 mm, width 0.32 mm, and height 0.31 mm) representonly a few percent (˜2%) of the overall size (˜30 mm×˜20 mm) of thewireless system. Experimentally, we observe that the system is robust tostretching up to ˜30% and beyond.

To study these effects quantitatively, we carried out full, threedimensional finite element simulations that examine the influence of thediode on the stretchability of the coil in the integrated system, as inFIG. 39 (top panel). Results in the bottom panels of FIG. 39, indicatethat: (1) The decrease in stretchability is modest, from 32.6% to 32.3%,when the diode is included and (2) The strain in the diode (i.e. theepoxy) is very small (<0.15%, much smaller than the strain needed toinduce fracture), even when the overall system is stretched by 32.3%.

The normal interface strain is also important. FIG. 24A shows thedistributions of maximum principal strains in a large Ecoflex substratewith a diode mounted in its center, for stretching to 30%. FIG. 40Billustrates the distribution of the substrate normal strain at thediode/substrate interface. The normal interface strain in this case isnegative, corresponding to compressive strain at the interface. Thisoutcome, which is consistent with theoretical predictions based onenergy release rates at an interface crack, minimizes the propensity fordelamination.

Coil Resistance Effect on the Wireless Power Transfer Efficiency.

The coil resistance/qualify factor is a critical parameter that dictatesthe efficiency. In additional experiments to illustrate the effects, wereplaced the 600 nm thick gold serpentine coils with otherwise similarones fabricated with copper at increased thicknesses. The results showthat coils formed using a 7 μm thick copper film (Dupont) have totalresistances of 185Ω, and generate a received power of 30.8 mW with aninput power of 179 mW (at a distance of 1 mm, similar to the previouslyreported results). The corresponding efficiency is 17.2%, whichrepresents more than a factor of three improvement over the original,gold coils (4.9%). Further reducing the coil resistance to 38Ω by using18 μm thick copper foil (MTI Corp.) improves the received power to 36.2mW, and the efficiency to 20.2%. See FIG. 43A.

These increases in thickness, and therefore power transfer efficiencies,lead to changes in the essential mechanics associated with stretching.In particular, as the thickness increases, the stretchability decreases.The coil with thickness of 7 μm offers a good balance between efficiencyand mechanical deformability, with ability to accommodate strains of˜25%. Images at various levels of strain appear in FIG. 43B, which agreewell with the finite element analysis results in terms of both themaximum uniaxial strain and the geometry of the coil serpentines (FIG.43C).

Modeling of the Charging Current in the Wireless Power TransmissionCircuit

The charging circuit can be described using the model below:

$\begin{matrix}{U_{0} = {{U(t)} + {L\frac{d\;{I(t)}}{dt}} + {{I(t)}R}}} & (1)\end{matrix}$Here U₀ is the charging source of 3 volts voltage. L and R are theassociated inductance and the resistance of the circuit. U(t) is thetime dependent readout of the voltmeter and I(t) is the time dependentreadout of an ammeter.

A program has been developed to simulate the I-V curve based on thepartial differential equation (1). The simulated time dependent currentI_(s)(t) based on U(t) is compared with measured I(t), and the resultsare shown in FIGS. 44A and 44B.

The program used to simulate the current:

#include<iostream.h> #include<stdio.h> #include<stdlib.h>#include<math.h> #include<time.h> #define tim 3500 #define start 265float curre[tim]; float nihe[tim]; float test[tim]; float voll[tim];FILE *fp; int main( ) {    int i,j,k;  fp=fopen(“Cur.txt”,“r”);   for(i=0;i<tim;i++)       {          fscanf(fp,“%f”,&curre[i]);         curre[i]=−curre[i];          cout<<curre[i]<<endl;       }   fclose(fp);    fp=fopen(“Vol.txt”,“r”);    for(i=0;i<tim;i++)       {         fscanf(fp,“%f”,&voll[i]);          cout<<voll[i]<<endl;       }   fclose(fp);    double coef1;    double coef2;   coef2=curre[1]/(voll[1]−voll[0]);    cout<<coef2<<endl;   test[0]=curre[0];    test[start]=curre[start];    double coef1th;   double maxh=1000000000;    double coef2th;    for(coef2=0;coef2<0.04;coef2=coef2+0.0001)    {       cout<<coef2<<endl;       for(coef1=0.94;coef1<=0.96; coef1=coef1+0.0001)       {       // cout<<coef1<<endl;      for(i=start+1;i<tim;i++)       {         test[i]=coef1*test[i−1]+(3−(voll[i]+voll[i−1])*         0.5)*coef2;       }       double poi=0;      for(i=start;i<tim;i++)       {         poi+=(test[i]−curre[i])*(test[i]−curre[i]);       }      if(poi<maxh)       {          coef1th=coef1;         coef2th=coef2;          maxh=poi;          for(intj=0;j<tim;j++)          {             nihe[j]=test[j];          }      }       }    }    cout<<coef1th<<endl;    cout<<coef2th<<endl;   cout<<maxh<<endl;    fp=fopen(“nihe.txt”,“w”);    for(i=0;i<tim;i++)   {       fprintf(fp, “%f”, −nihe[i]);       fprintf(fp, “\n”);    }   fclose(fp);    fp=fopen(“canshu.txt”,“w”);    fprintf(fp, “%f”,coef1th);    fprintf(fp, “\n”);    fprintf(fp, “%f”, coef2th);   fprintf(fp, “\n”);    fclose(fp);    return(1); }

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Example 3: Mechanics of Ultra-Stretchable Self-Similar SerpentineInterconnects

Abstract

Electrical interconnects that adopt self-similar, serpentine layoutsoffer exceptional levels of stretchability in systems that consist ofcollections of small, non-stretchable active devices, in the so-calledisland-bridge design. This Example develops analytical models offlexibility and elastic stretchability for such structures andestablishes recursive formulae at different orders of self-similarity.The analytic solutions agree well with finite element analysis (FEA),with both demonstrating that the elastic stretchability more thandoubles when the order of the self-similar structure increases by one.Design optimization yields 90% and 50% elastic stretchability forsystems with surface filling ratios of 50% and 70% of active devices,respectively.

1. Introduction

Interest in development of electronic and optoelectronic systems thatoffer elastic response to large strain (>>1%) deformation has grownrapidly in recent years [1-10], due in part to a range of importantapplication possibilities that cannot be addressed with establishedtechnologies, such as wearable photovoltaics [11], ‘epidermal’health/wellness monitors [8], eyeball-like digital cameras [9,12], andsensitive robotic skins [13-15]. Many of these stretchable devices adoptthe island-bridge design [8,12,16-18], where the active components aredistributed in small, localized regions (i.e. islands) and are joined bynarrow, deformable electrical and/or mechanical interconnects (i.e.bridges). Under stretching conditions, the relatively stiff islandseffectively isolate the active components (usually brittle materials)from strains that could cause fracture (e.g., <1%); the bridgestructures accommodate nearly all of the deformation [17-19]. For manypractical devices, the island-bridge design must achieve simultaneouslytwo competing goals, i.e., high surface filling ratio of active devices,and high stretchability of the entire system. Demonstrated designsolutions involve either serpentine [1,8,17,20-27] or non-coplanar[12,18] interconnects. These technologies, however, typically givelevels of total stretchability that are less than 50%, in systems thatdo no significantly sacrifice areal coverage. Recently, Xu et al. [19]illustrated an alternative type of interconnect design that exploitsself-similar serpentine geometries (shown in FIG. 45a ), a type ofspace-filling curve. This concept enabled lithium-ion batteries with thebiaxial stretchability up to ˜300%, and areal coverages of activematerials as high as ˜50%. Comprehensive experimental and numericalinvestigations indicated that such self-similar serpentine interconnectspossess improved levels of stretchability compared to traditionalserpentine structures for a given spacing between adjacent islands. Thenature of the space-filling geometry in these structures and themechanisms for their ordered unraveling were found to play importantroles.

This Example aims at developing an analytic model to study theflexibility and elastic stretchability (referred to simply asstretchability in the following) of self-similar serpentineinterconnects, and to establish the design guidelines for optimizing thestretching limit. Here, we focus on the scenario that the interconnectsare not bonded to the supporting substrate such that deformation canoccur freely and the interactions with the substrate can be neglected.Such freely suspended interconnects can be fabricated by either of twomethods: i) molding surface relief structures on the elastomericsubstrate [16,18,28], and bonding the islands onto the top of the raisedrelief; ii) designing the mask of SiO₂ deposition to enable selectivebonding of the islands onto the substrate [29,30], while leaving theinterconnects with a minimum interaction with the substrate. The presentstudy mainly focuses on relative thick interconnects with the thicknesscomparable to the width, as required for applications that demand lowelectrical resistance, such as wireless inductive coils [19], andphotovoltaic modules [11]. In such cases, the deformation of theinterconnects is governed by in-plane bending, rather than buckling,when the system is under stretching. Here, the critical buckling strainis large compared to the stretchability [31], such that buckling is nottriggered within the range of acceptable deformations. This mechanics isqualitatively different from that of the types of free-standing, thinserpentine interconnects that have been investigated previously[17,31-33]. For free-standing, thick self-similar interconnects,analytic models of the flexibility and stretchability are established inthis study. The models are then extended to arbitrary self-similarorders. The results establish design guidelines for practicalapplications.

This Example is outlined as follows: Section 2 focuses on the simplestgeometric configuration, self-similar rectangular interconnects, toillustrate the mechanics model for analyzing the flexibility andstretchability. The analytic model is extended to generalizedself-similar rectangular and serpentine interconnects in Section 3. Thestretchability of self-similar interconnects is studied in Section 4.Section 5 presents the optimal design of self-similar serpentineinterconnects for stretchable electronics to illustrate its advantage inachieving high system stretchability.

2. Self-Similar Rectangular Interconnects

This section focuses on a geometrically simple self-similar interconnectin a rectangular configuration (as shown in FIG. 45b ), to illustrateits structure, flexibility and stretchability. The rectangularinterconnect is a variant of the traditional serpentine interconnect(top panel of FIG. 45c ), and is convenient for constructingself-similar structures because of its simple geometry. To determine theflexibility of self-similar rectangular interconnects, the key is toestablish the relation between the flexibility of neighboring orders,i.e., the recursion formula. We first take the 1^(st) order self-similarrectangular interconnect as an example to illustrate the model as inSection 2.2, and then generalize the theoretical framework to the 2^(nd)order and arbitrary order in Sections 2.3 and 2.4, respectively.

2.1. Geometry

This subsection introduces the geometric construction of self-similarrectangular interconnects. The 1^(st) order (original) rectangularinterconnect consists of two sets of straight wires that areperpendicular to each other and connected in series, as shown in theblack box of FIG. 45b . The 2^(nd) order rectangular interconnect, shownin the blue box of FIG. 45b , is created by reducing the scale of the1^(st) order interconnect, rotating the structure by 90°, and thenconnecting multiple copies of it in a fashion that reproduces the layoutof original geometry. The wide blue line in FIG. 45b represents the2^(nd) order geometry that is similar to the 1^(st) order rectangulargeometry. By implementing the same algorithm, we can generate the 3^(rd)and 4^(th) order rectangular interconnects, as illustrated in the redand purple boxes of FIG. 45b , where the red and purple lines denote the3^(rd) and 4^(th) order geometries, respectively.

For self-similar rectangular interconnects, let m denote the number ofunit cell and η the height/spacing aspect ratio at each order. Thereforethe lengths of horizontal and vertical lines of the i^(th) order (i=1 .. . n), l^((i)) and h^((i)) (FIG. 45b ), are related byh ^((i)) =ηl ^((i)).  (1)In addition, the height of i^(th) order geometry equals to the distancebetween two ends of (i−1)^(th) order geometry, that ish ^((i))=2 ml^((i−1))(i=2 . . . n).  (2)Equations (1) and (2) give the length and height at any order in termsof l^((n)), η and m, asl ^((i))=(η/2m)^(n−i) l ^((n)) ,h ^((i))=η(η/2m)^(n−i) l ^((n)),(i=1 . .. n).  (3)

This indicates that the geometry of an arbitrary self-similarrectangular interconnect is characterized by one base length (l^((n)))and three non-dimensional parameters, namely the self-similar order (n),the height/spacing ratio (η) and number (m) of unit cell. It should bementioned that, for n≥3, there is an additional constraint on theheight/spacing ratio η because of the following relation, which can beobserved from the geometry of 3^(rd) order rectangular interconnectshown in FIGS. 45b and 47b (to be discussed),l ^((i))=(2m _(h)+1)l(i−2)(i=3 . . . n),  (4)where m_(h) is the number of full unit cells in the structurerepresented by the horizontal part of the i^(th) order geometry (i=3 . .. n). Equations (3) and (4) give the constraint on the height/spacingratio η for n≥3

$\begin{matrix}{{\eta = {\frac{2m}{\sqrt{{2\; m_{h}} + 1}}\mspace{14mu}\left( {i = {3\mspace{14mu}\ldots\mspace{14mu} n}} \right)}},} & (5)\end{matrix}$i.e., the height/spacing ratio can only take some discrete values forn≥3. FIG. 45b shows a set of self-similar rectangular interconnects,from n=1 to 4, with m=4 and η=8/√{square root over (11)}.2.2. Flexibility of 1^(st) Order Rectangular Interconnects

FIG. 46a shows a schematic illustration of the 1^(st) order self-similarrectangular interconnect with m unit cells and height/spacing ratio η.As illustrated in FIG. 46b , a representative unit cell (e.g., thek^(th) unit cell) of the 1^(st) order structure is composed of fivestraight wires (i.e., 0^(th) order structure) (Parts I to V). Thevertical wires, Parts I and III, have a length of h⁽¹⁾/2, and Part IIhas a length of h⁽¹⁾. The horizontal wires, Parts IV and V, have alength of l⁽¹⁾.

Consider the 1^(st) order rectangular interconnect clamped at the leftend, and subject to an axial force N (along the direction between thetwo ends of the interconnect), a shear force Q (normal to N), and abending moment M, at the right end, within the plane of interconnect, asshown in FIG. 46a . The width (w) and thickness (t) of the serpentineinterconnect are usually much smaller than the length such that thestructure can be modeled as a curved beam. Let u and v denote thedisplacements at the right end, along and normal to the axial directionof the interconnect (parallel to N and to Q), respectively, and θ is therotation angle (FIG. 46a ). They are related to (N, Q, M) via the strainenergy W⁽¹⁾ in the interconnect by

$\begin{matrix}{{\begin{pmatrix}u \\v \\\theta\end{pmatrix} = {\begin{bmatrix}{{\partial W^{(1)}}/{\partial N}} \\{{\partial W^{(1)}}/{\partial Q}} \\{{\partial W^{(1)}}/{\partial M}}\end{bmatrix} = {{\begin{bmatrix}T_{11}^{(1)} & T_{12}^{(1)} & T_{13}^{(1)} \\T_{12}^{(1)} & T_{22}^{(1)} & T_{23}^{(1)} \\T_{13}^{(1)} & T_{23}^{(1)} & T_{33}^{(1)}\end{bmatrix}\begin{pmatrix}N \\Q \\M\end{pmatrix}} = {T^{(1)}\begin{pmatrix}N \\Q \\M\end{pmatrix}}}}},} & (6)\end{matrix}$

where w⁽¹⁾═(N,Q,M)T⁽¹⁾ (N,Q,M)^(T)/2 is a quadratic function of N, Q,and M for linear elastic behavior of the interconnect; and T⁽¹⁾ is thesymmetric flexibility matrix of the 1^(st) order interconnect and is tobe determined. The strain energy also equals the sum of strain energyW⁽⁰⁾ in all 0^(th) order interconnects (Parts I to V), i.e.

$\begin{matrix}{{W^{(1)} = {W^{(0)} = {\sum\limits_{k = 1}^{m}\;\left( {W_{k}^{I} + W_{k}^{II} + W_{k}^{III} + W_{k}^{IV} + W_{k}^{V}} \right)}}},} & (7)\end{matrix}$where W_(k) ^(I) to W_(k) ^(V) represent the strain energy of eachcomponent in the k^(th) unit cell. For the 0^(th) order structure, i.e.a straight wire with length l and bending stiffness El, the beam theorygives the flexibility matrix as [34]

$\begin{matrix}{{T^{(0)}(l)} = {\frac{1}{6{EI}}{\begin{pmatrix}0 & 0 & 0 \\0 & {2\; l^{\; 3}} & {3\; l^{\; 2}} \\0 & {3l^{\; 2}} & {6\; l}\end{pmatrix}.}}} & (8)\end{matrix}$Here the membrane energy is neglected. The free body diagram of thek^(th) unit cell of the 1^(st) order interconnect (FIG. 46b ) gives theaxial force, shear force and bending moment in each wire, and the strainenergy of each 0^(th) order interconnect can then be obtained as

$\begin{matrix}{{\begin{pmatrix}W_{k}^{I} \\W_{k}^{II} \\W_{k}^{III} \\W_{k}^{IV} \\W_{k}^{V}\end{pmatrix} = {\frac{1}{2}\left( {N,Q,M} \right)\begin{Bmatrix}{D_{I}{T^{(0)}\left\lbrack {h^{(1)}/2} \right\rbrack}D_{I}^{T}} \\{D_{II}{T^{(0)}\left\lbrack h^{(1)} \right\rbrack}D_{II}^{T}} \\{D_{III}{T^{(0)}\left\lbrack {h^{(1)}/2} \right\rbrack}D_{III}^{T}} \\{D_{IV}{T^{(0)}\left\lbrack l^{(1)} \right\rbrack}D_{IV}^{T}} \\{D_{V}{T^{(0)}\left\lbrack l^{(1)} \right\rbrack}D_{V}^{T}}\end{Bmatrix}\left( {N,Q,M} \right)^{T}}}{{{{where}\mspace{14mu} D_{I}} = \begin{bmatrix}0 & 1 & {{- h^{(1)}}/2} \\1 & 0 & {{- \left( {{2m} - {2k} + 2} \right)}l^{(1)}} \\0 & 0 & {- 1}\end{bmatrix}},{D_{II} = \begin{bmatrix}0 & 1 & {{- h^{(1)}}/2} \\{- 1} & 0 & {\left( {{2m} - {2k} + 2} \right)l^{(1)}} \\0 & 0 & 1\end{bmatrix}},}} & (9) \\{{{D_{III} = \begin{bmatrix}0 & 1 & 0 \\1 & 0 & {{- \left( {{2m} - {2k}} \right)}l^{(1)}} \\0 & 0 & {- 1}\end{bmatrix}},{D_{IV} = \begin{bmatrix}1 & 0 & {h^{(1)}/2} \\0 & 1 & {\left( {{2m} - {2k} + 1} \right)l^{(1)}} \\0 & 0 & 1\end{bmatrix}},\;{and}}{D_{V} = {\begin{bmatrix}1 & 0 & {h^{(1)}/2} \\0 & {- 1} & {{- \left( {{2m} - {2k}} \right)}l^{(1)}} \\0 & 0 & {- 1}\end{bmatrix}.}}} & (10)\end{matrix}$Substitution of Eq. (9) into Eq. (7) gives the recursive formula betweenthe flexibility matrices of 1^(st) and 0^(th) order interconnects as

$\begin{matrix}{T^{(1)} = {\sum\limits_{k = 1}^{m}\;{\begin{Bmatrix}\begin{matrix}{{D_{I}{T^{(0)}\left\lbrack {h^{(1)}/2} \right\rbrack}D_{I}^{T}} + {D_{II}{T^{(0)}\left\lbrack h^{(1)} \right\rbrack}D_{II}^{T}} +} \\{{D_{III}{T^{(0)}\left\lbrack {h^{(1)}/2} \right\rbrack}D_{III}^{T}} +}\end{matrix} \\{{D_{IV}{T^{(0)}\left\lbrack l^{(1)} \right\rbrack}D_{IV}^{T}} + {D_{V}{T^{(0)}\left\lbrack l^{(1)} \right\rbrack}D_{V}^{T}}}\end{Bmatrix}.}}} & (11)\end{matrix}$Substitution of T⁽⁰⁾ in Eq. (8) into the above equation gives a simpleexpression of the flexibility of 1^(st) order interconnect in terms ofthe number of unit cells m, height/spacing ratio η and l⁽¹⁾,

$\begin{matrix}{{T^{(1)}\left\lbrack {m,\eta,l^{(1)}} \right\rbrack} = {\frac{1}{EI}{\begin{Bmatrix}{m{\frac{\eta^{3} + {3\eta^{2}}}{6}\left\lbrack l^{(1)} \right\rbrack}^{3}} & {m{\frac{\eta\left( {\eta + 2} \right)}{4}\left\lbrack l^{(1)} \right\rbrack}^{3}} & 0 \\{m{\frac{\eta\left( {\eta + 2} \right)}{4}\left\lbrack l^{(1)} \right\rbrack}^{3}} & {\frac{{\left( {{8m^{3}} + m} \right)\eta} + {8m^{3}}}{3}\left\lbrack l^{(1)} \right\rbrack}^{3} & {2{{m^{2}\left( {\eta + 1} \right)}\left\lbrack l^{(1)} \right\rbrack}^{2}} \\0 & {2{{m^{2}\left( {\eta + 1} \right)}\left\lbrack l^{(1)} \right\rbrack}^{2}} & {2{m\left( {\eta + 1} \right)}l^{(1)}}\end{Bmatrix}.}}} & (12)\end{matrix}$

For the convenience of generalization to higher order (n) structure, thefollowing non-dimensional form of flexibility matrix is adopted

$\begin{matrix}{{\begin{bmatrix}{u/l^{(i)}} \\{v/l^{(i)}} \\\theta\end{bmatrix} = {\frac{l^{(i)}}{EI}{{\overset{\_}{T}}^{(i)}\begin{bmatrix}{Nl}^{(i)} \\{Ql}^{(i)} \\M\end{bmatrix}}}},{i = {1\mspace{14mu}\ldots\mspace{14mu} n}},} & (13)\end{matrix}$where T ^((i)) is dimensionless, and T ⁽¹⁾ is given by

$\begin{matrix}{{{\overset{\_}{T}}^{(1)}\left( {m,\eta} \right)} = {\begin{bmatrix}{m\frac{\eta^{3} + {3\eta^{2}}}{6}} & {m\frac{\eta\left( {\eta + 2} \right)}{4}} & 0 \\{m\frac{\eta\left( {\eta + 2} \right)}{4}} & \frac{{\left( {{8m^{3}} + m} \right)\eta} + {8m^{3}}}{3} & {2{m^{2}\left( {\eta + 1} \right)}} \\0 & {2{m^{2}\left( {\eta + 1} \right)}} & {2{m\left( {\eta + 1} \right)}}\end{bmatrix}.}} & (14)\end{matrix}$For the 0^(th) order structure, i.e., a straight wire of length λ, thenon-dimensional flexibility matrix is defined as (u/λ,v/λ,θ)^(T)=(λ/EI)T⁽⁰⁾ (Nλ,Qλ,M)^(T), where

$\begin{matrix}{{\overset{\_}{T}}^{(0)} = {\begin{pmatrix}0 & 0 & 0 \\0 & \frac{1}{3} & \frac{1}{2} \\0 & \frac{1}{2} & 1\end{pmatrix}.}} & (15)\end{matrix}$2.3. Flexibility of 2^(nd) Order Rectangular Interconnect

The recursive formula for the flexibility matrix of 2^(nd) orderinterconnect is established in this section. A representative unit cellof the 2^(nd) order structure is composed of three 1^(st) orderstructures (Parts I to III), and two straight wires (i.e., 0^(th) orderstructure) (Parts IV and V) with length of l⁽²⁾, as illustrated in FIG.47a . The 1^(st) order structures, Parts I or III, consist of m/2 (m isan even integer) unit cells, and Part II consists of m unit cells.

The strain energy of the 2^(nd) order structure can be expressed interms of the dimensionless flexibility matrix as

$\begin{matrix}\begin{matrix}{W^{(2)} = {{\frac{l^{(2)}}{2\;{EI}}\left\lbrack {{Nl}^{(2)},{Ql}^{(2)},M} \right\rbrack}{{\overset{\_}{T}}^{(2)}\left\lbrack {{Nl}^{(2)},{Ql}^{(2)},M} \right\rbrack}^{T}}} \\{= {\frac{l^{(2)}}{2\;{EI}}{\left( {N,Q,M} \right)\begin{bmatrix}l^{(2)} & 0 & 0 \\0 & l^{(2)} & 0 \\0 & 0 & 1\end{bmatrix}}}} \\{{{{\overset{\_}{T}}^{(2)}\begin{bmatrix}l^{(2)} & 0 & 0 \\0 & l^{(2)} & 0 \\0 & 0 & 1\end{bmatrix}}\left\lbrack {N,Q,M} \right\rbrack}^{T},}\end{matrix} & (16)\end{matrix}$where T ⁽²⁾ is to be determined. The strain energy also equals the sumof strain energy in all 1^(st) order (Parts I to III, FIG. 47a ) and0^(th) order (Parts IV and V, FIG. 47a ) interconnects, i.e.

$\begin{matrix}{{W^{(2)} = {\sum\limits_{k = 1}^{m}\;\left( {W_{k}^{I} + W_{k}^{II} + W_{k}^{III} + W_{k}^{IV} + W_{k}^{V}} \right)}},} & (17) \\{where} & \; \\{W_{k}^{II} = {{\frac{l^{(1)}}{2\;{EI}}\left\lbrack {{Nl}^{(1)},{Ql}^{(1)},M} \right\rbrack}{\overset{\_}{D}}_{II}{{\overset{\_}{T}}^{(1)}\left( {m,\eta} \right)}{{\overset{\_}{D}}_{II}^{T}\left\lbrack {{Nl}^{(1)},{Ql}^{(1)},M} \right\rbrack}^{T}}} & (18)\end{matrix}$is the strain energy in Part II (1^(st) order structure, m unit cell)with

${\overset{\_}{D}}_{II} = \begin{bmatrix}0 & 0 & {- m} \\{- 1} & 0 & {\left( {{4m} - {4k} + 2} \right)m\;\eta^{- 1}} \\0 & 0 & 1\end{bmatrix}$being the normalized D_(II) in Eq. (10) (with l⁽¹⁾ and h⁽¹⁾ replaced byl⁽²⁾ and h⁽²⁾, respectively);

$\begin{matrix}{{W_{k}^{IV} = {{\frac{l^{(2)}}{2\;{EI}}\left\lbrack {{Nl}^{(2)},{Q\; l^{(2)}},M} \right\rbrack}{\overset{\_}{D}}_{IV}{\overset{\_}{T}}^{(0)}{{\overset{\_}{D}}_{IV}^{T}\left\lbrack {{Nl}^{(2)},{Q\; l^{(2)}},M} \right\rbrack}^{T}}}{W_{k}^{V} = {{\frac{l^{(2)}}{2\;{EI}}\left\lbrack {{Nl}^{(2)},{Q\; l^{(2)}},M} \right\rbrack}{\overset{\_}{D}}_{V}{\overset{\_}{T}}^{(0)}{{\overset{\_}{D}}_{V}^{T}\left\lbrack {{Nl}^{(2)},{Q\; l^{(2)}},M} \right\rbrack}^{T}}}} & (19)\end{matrix}$are the strain energy in Parts IV and V [0^(th) order structure, lengthλ=l⁽²⁾] with

${\overset{\_}{D}}_{IV} = {\begin{pmatrix}1 & 0 & \frac{\eta}{2} \\0 & 1 & {{2\; m} - {2\; k} + 1} \\0 & 0 & 1\end{pmatrix}{\mspace{11mu}\;}{and}}$${\overset{\_}{D}}_{V} = \begin{pmatrix}1 & 0 & \frac{\eta}{2} \\0 & {- 1} & {{{- 2}\; m} + {2\; k}} \\0 & 0 & {- 1}\end{pmatrix}$being the normalized D_(IV) and D_(V) in Eq. (10) [with l⁽¹⁾ and h⁽¹⁾replaced by l⁽²⁾ and h⁽²⁾, respectively];

$\begin{matrix}{{W_{k}^{I} = {{\frac{l^{(2)}}{2\;{EI}}\left\lbrack {{Nl}^{(1)},{Q\; l^{(1)}},M} \right\rbrack}{\overset{\_}{D}}_{I}{{\overset{\_}{T}}^{(1)}\left( {\frac{m}{2},\eta} \right)}{{\overset{\_}{D}}_{I}^{T}\left\lbrack {{Nl}^{(1)},{Q\; l^{(1)}},M} \right\rbrack}^{T}}}{W_{k}^{III} = {{\frac{l^{(1)}}{2\;{EI}}\left\lbrack {{Nl}^{(1)},{Q\; l^{(1)}},M} \right\rbrack}{\overset{\_}{D}}_{III}{{\overset{\_}{T}}^{(1)}\left( {\frac{m}{2},\eta} \right)}{{\overset{\_}{D}}_{III}^{T}\left\lbrack {{Nl}^{(1)},{Q\; l^{(1)}},M} \right\rbrack}^{T}}}} & (20)\end{matrix}$are the strain energy in Parts I and III (1^(st) order structure, m/2unit cell) with

${\overset{\_}{D}}_{I} = {\begin{bmatrix}0 & 1 & {- m} \\1 & 0 & {{- 4}\left( {m - k + 1} \right)m\;\eta^{- 1}} \\0 & 0 & {- 1}\end{bmatrix}{\mspace{11mu}\;}{and}}$${\overset{\_}{D}}_{III} = \begin{bmatrix}0 & 1 & 0 \\1 & 0 & {{- 4}\left( {m - k} \right)m\;\eta^{- 1}} \\0 & 0 & {- 1}\end{bmatrix}$being the normalized D_(I) and D_(III) in Eq. (10) [with l⁽¹⁾ and h⁽¹⁾replaced by l⁽²⁾ and h⁽²⁾, respectively].

Substitution of Eqs. (18)-(20) into Eq. (17) gives the recursive formulafor the flexibility matrix of 2^(nd) order interconnect as

$\begin{matrix}{{{\overset{\_}{T}}^{(2)} = {{\frac{\eta}{2\; m}\begin{pmatrix}\frac{\eta}{2\; m} & 0 & 0 \\0 & \frac{\eta}{2\; m} & 0 \\0 & 0 & 1\end{pmatrix}{\sum\limits_{k = 1}^{m}\;{\begin{Bmatrix}{{{{\overset{\_}{D}}_{I}\left\lbrack {{{\overset{\_}{T}}^{(1)}{K\left( {m\;\eta} \right)}} + {{K^{T}\left( {m\;\eta} \right)}{\overset{\_}{T}}^{(1)}}} \right\rbrack}{\overset{\_}{D}}_{I}^{T}} +} \\{{{\overset{\_}{D}}_{II}{\overset{\_}{T}}^{(1)}{\overset{\_}{D}}_{II}^{T}} +} \\{{{\overset{\_}{D}}_{III}\left\lbrack {{{\overset{\_}{T}}^{(1)}{K\left( {m\;\eta} \right)}} + {{K^{T}\left( {m\;\eta} \right)}{\overset{\_}{T}}^{(1)}}} \right\rbrack}{\overset{\_}{D}}_{III}^{T}}\end{Bmatrix}\begin{pmatrix}\frac{\eta}{2\; m} & 0 & 0 \\0 & \frac{\eta}{2\; m} & 0 \\0 & 0 & 1\end{pmatrix}}}} + {\sum\limits_{k = 1}^{m}\left\lbrack {{{\overset{\_}{D}}_{IV}{\overset{\_}{T}}^{(0)}{\overset{\_}{D}}_{IV}^{T}} + {{\overset{\_}{D}}_{V}{\overset{\_}{T}}^{(0)}{\overset{\_}{D}}_{V}^{T}}} \right\rbrack}}},} & (21) \\{\mspace{79mu}{where}} & \; \\{\mspace{79mu}{{K\left( {m\;\eta} \right)} = {\frac{1}{4}\begin{pmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & {{- m}\;\eta} & 1\end{pmatrix}}}} & (22)\end{matrix}$results from the identity

$\begin{matrix}{{{\overset{\_}{T}}^{(1)}\left( {\frac{m}{2},\eta} \right)} = {{{{\overset{\_}{T}}^{(1)}\left( {m,\eta} \right)}{K\left( {m\;\eta} \right)}} + {{K^{T}\left( {m\;\eta} \right)}{{{\overset{\_}{T}}^{(1)}\left( {m,\eta} \right)}.}}}} & (23)\end{matrix}$

Substitution of T ⁽⁰⁾ and T ⁽¹⁾ in Eqs. (15) and (14) into Eq. (21)gives T ⁽²⁾ as

$\begin{matrix}{{{{{\overset{\_}{T}}^{(2)}\left( {m,\eta} \right)} = \left\lbrack \begin{matrix}{\eta^{2}\frac{\eta^{2} + {2\;{m^{2}\left( {f + 2} \right)}}}{12\; m}} & {\frac{m}{4}{\eta\left( {f + 1} \right)}} & 0 \\{\frac{m}{4}{\eta\left( {f + 1} \right)}} & \frac{{\eta^{5}\left( {\eta + 3} \right)} + {8{m^{2}\left( {f - 1} \right)}} + {64\; m^{4}f}}{24\; m} & {2\; m^{2}f} \\0 & {2\; m^{2}f} & {2\;{mf}}\end{matrix} \right\rbrack},{where}}{f = {\eta^{2} + \eta + 1.}}} & (24)\end{matrix}$2.4. Flexibility of Higher Order Rectangular Interconnect

For the higher order (n≥3) rectangular interconnect, a representativeunit cell is composed of three (n−1) order structures (Parts I to III),and two (n−2) order structures (Parts IV and V). The (n−1) orderstructures, Parts I or III, consist of m/2 (m is an even integer) unitcells, and Part II consists of m unit cells. The recursive formula (21)becomes¹

$\begin{matrix}{{{\overset{\_}{T}}^{(n)} = {{{\frac{\eta}{2\; m}\begin{pmatrix}\frac{\eta}{2\; m} & 0 & 0 \\0 & \frac{\eta}{2\; m} & 0 \\0 & 0 & 1\end{pmatrix}{\sum\limits_{k = 1}^{m}\;{\begin{Bmatrix}{{{{\overset{\_}{D}}_{I}\left\lbrack {{{\overset{\_}{T}}^{({n - 1})}{K\left( {m\;\eta} \right)}} + {{K^{T}\left( {m\;\eta} \right)}{\overset{\_}{T}}^{({n - 1})}}} \right\rbrack}{\overset{\_}{D}}_{I}^{T}} +} \\{{{\overset{\_}{D}}_{II}{\overset{\_}{T}}^{({n - 1})}{\overset{\_}{D}}_{II}^{T}} +} \\{{{\overset{\_}{D}}_{III}\left\lbrack {{{\overset{\_}{T}}^{({n - 1})}{K\left( {m\;\eta} \right)}} + {{K^{T}\left( {m\;\eta} \right)}{\overset{\_}{T}}^{({n - 1})}}} \right\rbrack}{\overset{\_}{D}}_{III}^{T}}\end{Bmatrix}\begin{pmatrix}\frac{\eta}{2\; m} & 0 & 0 \\0 & \frac{\eta}{2\; m} & 0 \\0 & 0 & 1\end{pmatrix}}}} + {\frac{\eta^{2}}{4m^{2}}{\sum\limits_{k = 1}^{m}{\left\lbrack {{{\overset{\_}{D}}_{IV}^{*}{\overset{\_}{T}}^{({n - 2})}{\overset{\_}{D}}_{IV}^{*T}} + {{\overset{\_}{D}}_{V}^{*}{\overset{\_}{T}}^{({n - 2})}{\overset{\_}{D}}_{V}^{*T}}} \right\rbrack\mspace{14mu}{for}\mspace{14mu} n}}}} \geq 3}}\mspace{79mu}{where}\mspace{79mu}{{\overset{\_}{D}}_{IV}^{*} = \left\lbrack \begin{matrix}\frac{\eta^{2}}{\left( {4\; m^{2}} \right)} & 0 & \frac{\eta}{2} \\0 & \frac{\eta^{2}}{\left( {4\; m^{2}} \right)} & {{2\; m} - {2\; k} + 1} \\0 & 0 & 1\end{matrix} \right\rbrack}} & (25) \\{\mspace{79mu}{and}} & \; \\{\mspace{79mu}{{\overset{\_}{D}}_{V}^{*} = {\left\lbrack \begin{matrix}\frac{\eta^{2}}{\left( {4\; m^{2}} \right)} & 0 & \frac{\eta}{2} \\0 & \frac{- \eta^{2}}{\left( {4\; m^{2}} \right)} & {{{- 2}\; m} + {2\; k}} \\0 & 0 & {- 1}\end{matrix} \right\rbrack.}}} & \;\end{matrix}$

¹The (n−2)^(th) order structures (e.g., Parts IV and V in FIG. 47b forthe case of n=3) have (m_(h)+1/2) unit cells at the (n−2)^(th) ordergeometry. However, because the contribution of the (n−2)^(th) orderstructures to the overall flexibility is much smaller than that of(n−1)^(th) order structures, the dimensionless flexibility of Parts IVand V can be approximated by the self-similar (n−2)^(th) orderstructures with m unit cells, which, as to be shown by FEA, gives rathergood accuracy.

3. Generalized Self-Similar Interconnects

The analytic model for self-similar rectangular interconnects in Section2 is extended to generalized self-similar rectangular and serpentineinterconnects in this section.

3.1. Generalized Self-Similar Rectangular Interconnects

The generalized rectangular interconnect still exhibits the rectangularshape (shown in FIG. 48), but does not require the same height/spacingratio across different orders, nor the number of unit cell. Each ordermay have its own height/spacing ratio η^((i)) and number of unit cellm^((i)) (i=1 . . . n), where only m^((n)) can be an odd number, and m⁽¹⁾to m^((n)) must be even numbers. FIG. 48 illustrates a generalized3^(rd) order self-similar rectangular interconnect. For the n^(th) ordergeneralized self-similar rectangular interconnect, the geometricrelations (1)-(3) become

$\begin{matrix}{{h^{(i)} = {\eta^{(i)}l^{(i)}}},} & (26) \\{{h^{(i)} = {2\; m^{({i - 1})}{l^{({i - 1})}\left( {i = {2\mspace{14mu}\ldots\mspace{14mu} n}} \right)}}},} & (27) \\{{l^{(i)} = {\left\lbrack {\prod\limits_{k = 1}^{n - i}\frac{\eta^{({n - k + 1})}}{2\; m^{({n - k})}}}\; \right\rbrack l^{(n)}}},{h^{(i)} = {{\eta^{(i)}\left\lbrack {\prod\limits_{k = 1}^{n - i}\frac{\eta^{({n - k + 1})}}{2\; m^{({n - k})}}}\; \right\rbrack}l^{(n)}}},{\left( {i = {{1\mspace{14mu}\ldots\mspace{14mu} n} - 1}} \right).}} & (28)\end{matrix}$

The flexibility matrix T ⁽⁰⁾ in Eq. (15) remains the same, while m and ηin Eq. (14) for T ⁽¹⁾ need to be replaced by m⁽¹⁾ and η⁽¹⁾,respectively. The recursive formulae for T ⁽²⁾ in Eq. (21) and T ^((n))(n≥3) in Eq. (25) now become

$\begin{matrix}{{{{\overset{\_}{T}}^{(2)} = {{{\frac{\eta^{(2)}}{2\; m^{(1)}}\begin{bmatrix}\frac{\eta^{(2)}}{2\; m^{(1)}} & 0 & 0 \\0 & \frac{\eta^{(2)}}{2\; m^{(1)}} & 0 \\0 & 0 & 1\end{bmatrix}}{\sum\limits_{k = 1}^{m^{(2)}}{\left\langle \begin{matrix}{{{\overset{\_}{D}}_{I}^{(2)}\begin{Bmatrix}{{\overset{\_}{T}}^{(1)}{K\left\lbrack {\left( {m^{(1)}\eta^{(1)}} \right\rbrack +} \right.}} \\{K^{T}\left\lbrack {\left( {m^{(1)}\eta^{(1)}} \right\rbrack{\overset{\_}{T}}^{(1)}} \right.}\end{Bmatrix}{\overset{\_}{D}}_{I}^{{(2)}T}} +} \\{{{\overset{\_}{D}}_{II}^{(2)}{\overset{\_}{T}}^{(1)}{\overset{\_}{D}}_{II}^{{(2)}T}} +} \\{{\overset{\_}{D}}_{III}^{{(2)}T}\begin{Bmatrix}{{\overset{\_}{T}}^{(1)}{K\left\lbrack {\left( {m^{(1)}\eta^{(1)}} \right\rbrack +} \right.}} \\{K^{T}\left\lbrack {\left( {m^{(1)}\eta^{(1)}} \right\rbrack{\overset{\_}{T}}^{(1)}} \right.}\end{Bmatrix}{\overset{\_}{D}}_{III}^{{(2)}T}}\end{matrix} \right\rangle\left\lbrack \begin{matrix}\frac{\eta^{(2)}}{2\; m^{(1)}} & 0 & 0 \\0 & \frac{\eta^{(2)}}{2\; m^{(1)}} & 0 \\0 & 0 & 1\end{matrix} \right\rbrack}}} + {\overset{m^{(2)}}{\sum\limits_{k = 1}}\left\lbrack {{{\overset{\_}{D}}_{IV}^{(2)}{\overset{\_}{T}}^{(0)}{\overset{\_}{D}}_{IV}^{{(2)}T}} + {{\overset{\_}{D}}_{V}^{(2)}{\overset{\_}{T}}^{(0)}{\overset{\_}{D}}_{V}^{{(2)}T}}} \right\rbrack}}},}\mspace{14mu}} & (29) \\{{\overset{\_}{T}}^{(n)} = {\frac{\eta^{(n)}}{2\; m^{({n - 1})}}\begin{pmatrix}\frac{\eta^{(n)}}{2\; m^{({n - 1})}} & 0 & 0 \\0 & \frac{\eta^{(n)}}{2\; m^{({n - 1})}} & 0 \\0 & 0 & 1\end{pmatrix}{\sum\limits_{k = 1}^{m^{(n)}}{\left\langle \begin{matrix}{{{\overset{\_}{D}}_{I}^{(n)}\begin{Bmatrix}{{\overset{\_}{T}}^{({n - 1})}{K\left\lbrack {\left( {m^{({n - 1})}\eta^{({n - 1})}} \right\rbrack +} \right.}} \\{K^{T}\left\lbrack {\left( {m^{({n - 1})}\eta^{({n - 1})}} \right\rbrack{\overset{\_}{T}}^{({n - 1})}} \right.}\end{Bmatrix}{\overset{\_}{D}}_{I}^{{(n)}T}} +} \\{{{\overset{\_}{D}}_{II}^{(n)}{\overset{\_}{T}}^{({n - 1})}{\overset{\_}{D}}_{II}^{{(n)}T}} +} \\{{\overset{\_}{D}}_{III}^{(n)}\begin{Bmatrix}{{\overset{\_}{T}}^{({n - 1})}{K\left\lbrack {\left( {m^{({n - 1})}\eta^{({n - 1})}} \right\rbrack +} \right.}} \\{K^{T}\left\lbrack {\left( {m^{({n - 1})}\eta^{({n - 1})}} \right\rbrack{\overset{\_}{T}}^{({n - 1})}} \right.}\end{Bmatrix}{\overset{\_}{D}}_{III}^{{(n)}T}}\end{matrix} \right\rangle \cdot {\quad{\left\lbrack \begin{matrix}\frac{\eta^{(n)}}{2\; m^{({n - 1})}} & 0 & 0 \\0 & \frac{\eta^{(n)}}{2\; m^{({n - 1})}} & 0 \\0 & 0 & 1\end{matrix} \right\rbrack + {\frac{\eta^{(n)}\eta^{({n - 1})}}{4\; m^{({n - 1})}m^{({n - 2})}}{\sum\limits_{k = 1}^{m^{(n)}}{\left\lbrack {{{\overset{\_}{D}}_{IV}^{{(n)}*}{\overset{\_}{T}}^{({n - 2})}{\overset{\_}{D}}_{IV}^{{(n)}*T}} + {{\overset{\_}{D}}_{V}^{{(n)}*}{\overset{\_}{T}}^{({n - 2})}{\overset{\_}{D}}_{V}^{{(n)}*T}}} \right\rbrack\mspace{14mu}{for}}}}}\mspace{11mu}}}}}} & (30) \\{\mspace{79mu}{{n \geq 3},\mspace{79mu}{where}}} & \; \\{\mspace{79mu}{{{\overset{\_}{D}}_{I}^{(n)} = \begin{Bmatrix}0 & 1 & {- m^{({n - 1})}} \\1 & 0 & {{- {4\left\lbrack {m^{(n)} - k + 1} \right\rbrack}}{m^{({n - 1})}\left\lbrack \eta^{(n)} \right\rbrack}^{- 1}} \\0 & 0 & {- 1}\end{Bmatrix}},}} & \; \\{\mspace{79mu}{{\overset{\_}{D}}_{II}^{(n)} = {\begin{Bmatrix}0 & 1 & {- m^{({n - 1})}} \\{- 1} & 0 & {\left\lbrack {{4m^{(n)}} - {4k} + 2} \right\rbrack{m^{({n - 1})}\left\lbrack \eta^{(n)} \right\rbrack}^{- 1}} \\0 & 0 & 1\end{Bmatrix}\mspace{14mu}{and}}}} & \; \\{\mspace{85mu}{{{\overset{\_}{D}}_{III}^{(n)} = {{\begin{bmatrix}0 & 1 & 0 \\1 & 0 & {{- {4\left\lbrack {m^{(n)} - k} \right\rbrack}}{m^{({n - 1})}\left\lbrack \eta^{(n)} \right\rbrack}^{- 1}} \\0 & 0 & {- 1}\end{bmatrix}\mspace{14mu}{for}\mspace{14mu} n} \geq 2}},}} & \; \\{\mspace{79mu}{{{\overset{\_}{D}}_{IV}^{(2)} = \begin{bmatrix}1 & 0 & \frac{\eta^{(2)}}{2} \\0 & 1 & {{2m^{(n)}} - {2k} + 1} \\0 & 0 & 1\end{bmatrix}},}} & \; \\{\mspace{79mu}{{{\overset{\_}{D}}_{V}^{(2)} = \begin{bmatrix}1 & 0 & \frac{\eta^{(2)}}{2} \\0 & {- 1} & {{{- 2}m^{(2)}} + {2k}} \\0 & 0 & {- 1}\end{bmatrix}},{and}}} & \; \\{{\overset{\_}{D}}_{IV}^{*{(n)}} = {\begin{bmatrix}\frac{\eta^{({n - 1})}\eta^{(n)}}{\left\lbrack {4\; m^{({n - 2})}m^{({n - 1})}} \right\rbrack} & 0 & \frac{\eta^{(n)}}{2} \\0 & \frac{\eta^{({n - 1})}\eta^{(n)}}{\left\lbrack {4\; m^{({n - 2})}m^{({n - 1})}} \right\rbrack} & {{2m^{(n)}} - {2k}\; + 1} \\0 & 0 & 1\end{bmatrix}\mspace{14mu}{and}}} & \; \\{{{\overset{\_}{D}}_{V}^{*{(n)}} = {\begin{bmatrix}\frac{\eta^{({n - 1})}\eta^{(n)}}{\left\lbrack {4\; m^{({n - 2})}m^{({n - 1})}} \right\rbrack} & 0 & \frac{\eta^{(n)}}{2} \\0 & \frac{{- \eta^{({n - 1})}}\eta^{(n)}}{\left\lbrack {4\; m^{({n - 2})}m^{({n - 1})}} \right\rbrack} & {{{- 2}m^{(n)}} - {2k}\; + 1} \\0 & 0 & {- 1}\end{bmatrix}\mspace{14mu}{for}}}\;\mspace{79mu}{n \geq 3.}} & \;\end{matrix}$3.2. Generalized Self-Similar Serpentine Interconnects

FIGS. 45b and 45c show the generalized self-similar serpentineinterconnect, which replaces the sharp corners in the rectangularconfiguration by half circles, as in Xu et al.'s experiments [19]. The1^(st) order serpentine interconnect consists of straight wires [lengthh⁽¹⁾−l⁽¹⁾] connected by half circles [diameter l⁽¹⁾], as shown in theblack box of FIG. 45c . A representative unit cell of the 2^(nd) orderserpentine interconnect, as shown in the blue box of FIG. 45c , iscomposed of two (horizontal) straight wires of length l⁽²⁾ and three(vertical) 1^(st) order serpentine interconnects (two with lengthsh⁽²⁾/2 and one with length of h⁽²⁾). The flexibility matrix T ⁽⁰⁾ forstraight wires is still given in Eq. (15), and the flexibility matrix T⁽¹⁾ for the 1^(st)-order serpentine interconnect is obtained as [31]

$\begin{matrix}{{{\overset{\_}{T}}^{(1)} = {\frac{m^{(1)}}{24}\begin{Bmatrix}{{4\; g^{3}} + {6\;\pi\; g^{2}} + {24\; g} + {3\;\pi}} & {6\left( {g^{2} + {\pi\; g} + 2} \right)} & 0 \\{6\left( {g^{2} + {\pi\; g} + 2} \right)} & {{{32\left\lbrack m^{(1)} \right\rbrack}^{2}\left( {{2\; g} + \pi} \right)} + {8\; g} + \pi} & {48\;{m^{(1)}\left( {g + \pi} \right)}} \\0 & {48\;{m^{(1)}\left( {g + \pi} \right)}} & {24\left( {{2\; g} + \pi} \right)}\end{Bmatrix}}}\text{}{where}{g = {\eta^{(1)} - 1.}}} & (31)\end{matrix}$

The 2^(nd) to 4^(th) (and higher) order geometries all exhibit arectangular geometry (shown in FIG. 45c ), which indicates that,strictly speaking, the self-similarity only starts at the 2^(nd) orderinterconnects. Comparison of the self-similar serpentine structure (FIG.45c ) to the rectangular one (FIG. 45b ) suggests that only their 1^(st)order geometries are different. Therefore, the recursive formulae inEqs. (29) and (30) still hold for the self-similar serpentine structure.

Substitution of T ⁽⁰⁾ in Eq. (15) and T ⁽¹⁾ in Eq. (14) into Eq. (31)gives T ⁽²⁾ as

$\quad{\quad\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}{{{{{\overset{\_}{T}}^{(2)}\left( {m,\eta} \right)} = {\frac{m^{(2)}}{24}\begin{bmatrix}{{{6\left\lbrack \eta^{(2)} \right\rbrack}^{2}\left( {4 - p} \right)} + {{6\left\lbrack \frac{\eta^{(2)}}{m^{(1)}} \right\rbrack}^{3}{\overset{\_}{T}}_{22}^{(1)}}} & {3\;{\eta^{(2)}\left( {p + 2} \right)}} & 0 \\{3\;{\eta^{(2)}\left( {p + 2} \right)}} & {{{32\left\lbrack m^{(2)} \right\rbrack}^{2}p} + {4p} - 8 + {{6\left\lbrack \frac{\eta^{(2)}}{m^{(1)}} \right\rbrack}{\overset{\_}{T}}_{11}^{(1)}}} & {24m^{2}p} \\0 & {24m^{(2)}p} & {24p}\end{bmatrix}}},{where}}{p = {{\eta^{(2)}\left\lbrack {{2\;\eta^{(1)}} + \pi - 2} \right\rbrack} + 2.}}} & (32)\end{matrix} & \;\end{matrix} & \;\end{matrix} & \;\end{matrix} & \;\end{matrix}}$

FIGS. 49a and 49b show the components of non-dimensional flexibilityversus the order (n) for self-similar rectangular and serpentineinterconnects for the height/spacing ratio η=8/√{square root over (11)}and number of unit cell m=4. The rectangular interconnect is slightlysofter than the serpentine one. The analytic results are validated byFEA, which is also shown in FIGS. 49a and 49b , for copper interconnectwith the elastic modulus E_(Cu)=119 GPa and Poisson's ratio v_(Cu)=0.34.The component T₁₃ is always zero, and is therefore not shown. The otherfive flexibility components all increase with n, and are more thandoubled for each n increasing by 1. For n from 1 to 4, these componentsincrease by more than 17 times, indicating that the higher-orderinterconnect becomes much softer than the lower-order one.

4. Stretchability

The interconnect usually spans the space between two rigid deviceislands (e.g., in FIG. 45a ), corresponding to clamped boundaryconditions at the two ends. For stretching u₀ of the self-similarinterconnect (with n orders), the boundary conditions are u=u₀, v=0 andθ=0, and Eq. (13) then gives the reaction forces, N and Q, and bendingmoment M as

$\begin{matrix}{\begin{Bmatrix}N \\Q \\M\end{Bmatrix} = {\frac{EI}{\left\lbrack l^{(n)} \right\rbrack^{3}}\frac{u_{0}}{{{\overset{\_}{T}}_{11}^{(n)}{\overset{\_}{T}}_{22}^{(n)}{\overset{\_}{T}}_{33}^{(n)}} - {{\overset{\_}{T}}_{11}^{(n)}\left\lbrack {\overset{\_}{T}}_{23}^{(n)} \right\rbrack}^{2} - {{\overset{\_}{T}}_{33}^{(n)}\left\lbrack {\overset{\_}{T}}_{12}^{(n)} \right\rbrack}^{2}}\begin{Bmatrix}{\;\begin{matrix}{{{\overset{\_}{T}}_{22}^{(n)}{\overset{\_}{T}}_{33}^{(n)}} - \left\lbrack {\overset{\_}{T}}_{23}^{(n)} \right\rbrack^{2} -} \\{{\overset{\_}{T}}_{12}^{(n)}{\overset{\_}{T}}_{33}^{(n)}} \\{l^{(n)}{\overset{\_}{T}}_{12}^{(n)}{\overset{\_}{T}}_{23}^{(n)}}\end{matrix}} \\\;\end{Bmatrix}}} & (33)\end{matrix}$since T ₁₃ ^((n))=0. The maximum strains for the rectangular andserpentine configurations are analyzed separately in Section 4.1 and4.2. Since no experiment result is available regarding thestretchability of relative thick self-similar rectangular or serpentineinterconnects, we only compare the analytic results to the FEA resultsfor validation. The experiment measurement of the stretchability andcomparison to analytic results will be considered in our future work.4.1. Generalized Self-Similar Rectangular Interconnects

For the 1^(st) order rectangular interconnect, it can be shown that themaximum strain occurs at the third nearest corners from the loadingpoints, as illustrated in FIG. 66a , which is well supported by FEAresults. The maximum strain in the interconnect can be then obtainedaccurately as

$\begin{matrix}{ɛ_{\max} = {\frac{w\left\lbrack {{2\; M} + {Nh}^{(1)} + {2{Ql}^{(1)}}} \right\rbrack}{4\;{EI}}.}} & \left( {34a} \right)\end{matrix}$For higher order structures with n≥2, the maximum strain can be wellapproximated by

$\begin{matrix}{ɛ_{\max} \approx {\frac{w\left\lbrack {{2\; M} + {Nh}^{(n)} + {2{Ql}^{(n)}} - {Qh}^{({n - 1})}} \right\rbrack}{4\;{EI}}\mspace{14mu}{for}\mspace{14mu}{\left( {n \geq 2} \right).}}} & \left( {34b} \right)\end{matrix}$Based on the yield criterion ε_(max)=ε_(yield), where ε_(yield) is theyield strain of the interconnect material (e.g., 0.3% for copper [35]),the stretchability of the generalized self-similar rectangularinterconnect is obtained as

$\begin{matrix}{{ɛ_{stretchability}^{(1)} = {\frac{ɛ_{yield}\mspace{11mu} l^{(1)}}{w}\frac{\eta^{(1)}}{12}\frac{{{{16\left\lbrack m^{(1)} \right\rbrack}^{2}\left\lbrack {\eta^{(1)} + 1} \right\rbrack}\left\lbrack {\eta^{(1)} + 3} \right\rbrack} - \left\lbrack {\eta^{(1)} + 6} \right\rbrack^{2}}{{{4\left\lbrack m^{(1)} \right\rbrack}^{2}\left\lbrack {\eta^{(1)} + 1} \right\rbrack} + {3{m^{(1)}\left\lbrack {\eta^{(1)} + 2} \right\rbrack}} - \eta^{(1)} - 6}}},} & \left( {35a} \right) \\{{ɛ_{stretchability}^{(n)} = {\frac{ɛ_{yield}\mspace{11mu} l^{(n)}}{w}\frac{2}{m^{(n)}}{\frac{{{\overset{\_}{T}}_{11}^{(n)}\left\lbrack {\overset{\_}{T}}_{23}^{(n)} \right\rbrack}^{2} + {{\overset{\_}{T}}_{33}^{(n)}\left\lbrack {\overset{\_}{T}}_{12}^{(n)} \right\rbrack}^{2} - {{\overset{\_}{T}}_{11}^{(n)}{\overset{\_}{T}}_{22}^{(n)}{\overset{\_}{T}}_{33}^{(n)}}}{\begin{matrix}{{2{{\overset{\_}{T}}_{12}^{(n)}\left\lbrack {{\overset{\_}{T}}_{33}^{(n)} - {\overset{\_}{T}}_{23}^{(n)}} \right\rbrack}} +} \\{{\left\{ {\left\lbrack {\overset{\_}{T}}_{23}^{(n)} \right\rbrack^{2} - {{\overset{\_}{T}}_{22}^{(n)}{\overset{\_}{T}}_{33}^{(n)}}} \right\}\eta^{(n)}} - {\frac{\eta^{({n - 1})}\eta^{(n)}}{2m^{({n - 1})}}{\overset{\_}{T}}_{12}^{(n)}{\overset{\_}{T}}_{33}^{(n)}}}\end{matrix}}}}}\mspace{79mu}{\left( {{{for}\mspace{14mu} n} \geq 2} \right).}} & \left( {35b} \right)\end{matrix}$

When the applied strain is smaller than the stretchability, theinterconnect undergoes linear, reversible deformations, and no plasticdeformation would accumulate, such that the interconnect would notsuffer from plastic fatigue under cyclic loadings. Equations (35a) and(35b) show clearly that the stretchability is linearly proportional toε_(yield)l^((n))/w. Therefore, in order to enhance the stretchability,it is better to adopt a metallic material with high yield strength andrelative low elastic modulus to give a high yield strain, such as thenano-grained size copper, or transforming metal nanocomposites [36].

4.2. Generalized Self-Similar Serpentine Interconnects

For 1^(st) order serpentine interconnect, as shown in FIG. 66b , themaximum strain always occurs at the nearest or second nearest halfcircle from the two ends. Let φ (0≤φ≤π) represent the location of thishalf circle. The bending strain on the circle can be given by

$\begin{matrix}{{ɛ(\varphi)} = {\frac{w\left\{ {{2M} + {N\left\lbrack {h^{(1)} - l^{(1)}} \right\rbrack} + {3\;{Ql}^{(1)}} + {l^{(1)}\left( {{N\mspace{11mu}\sin\mspace{11mu}\varphi} - {Q\mspace{11mu}\cos\mspace{11mu}\varphi}} \right)}} \right\}}{4E\; I}.}} & (36)\end{matrix}$It reaches the maximum at φ=tan⁻¹(−N/Q), and the maximum strain is givenby

$\begin{matrix}{ɛ_{\max} = {\frac{w\left\{ {{2M} + {N\left\lbrack {h^{(1)} - l^{(1)}} \right\rbrack} + {3\;{Ql}^{(1)}} + {l^{(1)}\sqrt{N^{2} + Q^{2}}}} \right\}}{4E\; I}.}} & (37)\end{matrix}$The stretchability of 1^(st) order serpentine interconnect is thenobtained as [via Eq. (33)]

$\begin{matrix}{ɛ_{stretchability} = {\frac{2ɛ_{yield}\mspace{11mu} l^{(1)}}{m^{(1)}w}{\frac{{{\overset{\_}{T}}_{11}^{(1)}{\overset{\_}{T}}_{22}^{(1)}{\overset{\_}{T}}_{33}^{(1)}} - {{\overset{\_}{T}}_{11}^{(1)}\left\lbrack {\overset{\_}{T}}_{23}^{(1)} \right\rbrack}^{2} - {{\overset{\_}{T}}_{33}^{(1)}\left\lbrack {\overset{\_}{T}}_{12}^{(1)} \right\rbrack}^{2}}{\begin{matrix}{{2{\overset{\_}{T}}_{12}^{(1)}{\overset{\_}{T}}_{23}^{(1)}} - {3{\overset{\_}{T}}_{12}^{(1)}{\overset{\_}{T}}_{33}^{(1)}} + {\left\{ {{{\overset{\_}{T}}_{22}^{(1)}{\overset{\_}{T}}_{33}^{(1)}} - \left\lbrack {\overset{\_}{T}}_{23}^{(1)} \right\rbrack^{2}} \right\}\left( {\eta^{(1)} - 1} \right)} +} \\\sqrt{\left\{ {{{\overset{\_}{T}}_{22}^{(1)}{\overset{\_}{T}}_{33}^{(1)}} - \left\lbrack {\overset{\_}{T}}_{23}^{(1)} \right\rbrack^{2}} \right\}^{2} + {\left\lbrack {\overset{\_}{T}}_{12}^{(1)} \right\rbrack^{2}\left\lbrack {\overset{\_}{T}}_{33}^{(1)} \right\rbrack}^{2}}\end{matrix}}.}}} & (38)\end{matrix}$The normalized stretchability ε_(stretchability)w/[ε_(yield)l⁽¹⁾]depends only on the height/spacing ratio η⁽¹⁾ and number of unit cellm⁽¹⁾. It increases with both η⁽¹⁾ and m⁽¹⁾, as shown in FIG. 50, andsaturates to

$\begin{matrix}{ɛ_{stretchability} = {\frac{ɛ_{yield}\mspace{11mu} l^{(1)}}{w} \cdot \frac{{4\left\lbrack \eta^{(1)} \right\rbrack}^{3} + {6{\left( {\pi - 2} \right)\left\lbrack \eta^{(1)} \right\rbrack}^{2}} - {12\left( {\pi - 3} \right)\eta^{(1)}} + {9\pi} - 28}{12\eta^{(1)}}}} & (39)\end{matrix}$for m⁽¹⁾→∞ (also shown in FIG. 50).

For higher order (n≥2) serpentine interconnects, Eq. (35b), togetherwith the corresponding flexibility matrix T ⁽²⁾ in Eq. (32) and T ^((n))in Eq. (30) for serpentine interconnects, give an excellentapproximation to the stretchability as compared to the FEA shown in FIG.51.

FIG. 52 shows the normalized stretchability,ε_(stretchability)w/[ε_(yield)l⁽¹⁾], versus the order n for self-similarrectangular and serpentine interconnects, where the height/spacing ratioη=8/√{square root over (11)} and number of unit cell m=4 at differentorders. The stretchability is more than doubled for each n increasing by1, indicating the elastic limit of the interconnect can be well improvedby adopting higher order self-similar design. FIG. 52 also shows thatthe analytic model agrees very well with the FEA results.

The analytic models and FEA results above are all for infinitesimaldeformation. FIG. 67 shows the effect of finite deformation onstretchability (determined by FEA) is negligible for both 1^(st) and2^(nd) order serpentine interconnects, with various combinations ofgeometric parameters. Therefore, the analytic models above give goodestimations of the stretchability. In real fabrications, the microscaleself-similar serpentine interconnect may have imperfections due tolithography defects especially along the sidewalls of the lines, andsuch geometric imperfections will increase for decreased pattern size(i.e., metal width and rounding radius) that may occur when increasingthe self similar order. These geometric imperfections are not accountedfor in the present study.

5. Optimal Design of Self-Similar Serpentine Interconnects forStretchable Electronics

Two competing goals of stretchable electronics [19,37] are 1) highsurface filling ratio of active devices, which requires small spacingbetween the device islands (FIG. 53a ); and 2) large stretchability ofthe system, which demands large spacing between the device islands.Prior approaches based on buckling of straight or conventionalserpentine interconnects achieve ˜100% stretchability [17,18,28,30]. Thestretchability (ε_(stretchability) ^(system)) of the system is relatedto that (ε_(stretchability) ^(interconnect)) of the interconnect byε_(stretchability) ^(system)=(ε_(stretchability)^(interconnect))(1−√{square root over (f)}),  (40)where f denotes the surface filling ratio. For ˜50% surface fillingratio of active devices, the ˜100% stretchability of the interconnecttranslates to ˜30% stretchability of the system, which is low for somebiomedical applications of stretchable electronics (to skin, heart, orelbow). The analytic models in Sections 3 and 4 can guide the design ofgeneralized self-similar interconnect to simultaneously achieve the twocompeting goals above.

The 2^(nd) order serpentine interconnects is studied to illustrate thedesign optimization in a square-shaped device island with arepresentative size H=1 mm and the surface filling ratio of 50% (FIG.53a ). The photolithography technology [38,39] for fabricating the metalinterconnect poses some constraints, such as the width w≤10 μm, roundingradius r_(rounding)≥10 μm, and the distance between neighboring arcs d≥5μm (FIG. 53a ). Other geometric parameters are optimized to achievelarge stretchability. FIG. 53b shows that the stretchability increaseswith the number of unit cells m⁽²⁾. The right panel of FIG. 53b show theoptimal design, which gives ˜308% stretchability of the interconnect,and corresponds to ˜90% stretchability of the system, outperforming theprevious designs using buckled interconnects [18,28]. Even for a muchlarger surface filling ratio 70%, Eq. (40) still gives ˜50%stretchability of the system.

6. Conclusions

This Example develops the analytic models of flexibility andstretchability for the self-similar interconnects. After thestraightforward design optimization, the analytic models, validated byFEA, show that the higher-order self-similar interconnect gives verylarge stretchability of the system, such as ˜90% for 50% surface fillingratio of active devices, or >50% stretchability for 70% surface fillingratio. The analytic models are useful for the development of stretchableelectronics that simultaneously demand large areal coverage of activedevices, such as stretchable photovoltaics [11] and electronic eye-ballcameras [12]. The concept of self-similar serpentine configuration canbe further combined with other strategies of stretchability enhancement,e.g., the control of wrinkling patterns, to give an enhanced level ofstretchability for interconnects bonded to the substrate.

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Example 4: A Hierarchical Computational Model for StretchableInterconnects with Fractal-Inspired Designs

Abstract

Stretchable electronics that require functional components with highareal coverages, antennas with small sizes and/or electrodes withinvisibility under magnetic resonance imaging can benefit from the useof electrical wiring constructs that adopt fractal inspired layouts. Dueto the complex and diverse microstructures inherent in high orderinterconnects/electrodes/antennas with such designs, traditionalnon-linear postbuckling analyses based on conventional finite elementanalyses (FEA) can be cumbersome and time-consuming. Here, we introducea hierarchical computational model (HCM) based on the mechanism ofordered unraveling for postbuckling analysis of fractal inspiredinterconnects, in designs previously referred to as ‘self-similar’,under stretching. The model reduces the computational efforts oftraditional approaches by many orders of magnitude, but with accuratepredictions, as validated by experiments and FEA. As the fractal orderincreases from 1 to 4, the elastic stretchability can be enhanced by˜200 times, clearly illustrating the advantage of simple concepts infractal design. These results, and the model in general, can beexploited in the development of optimal designs in wide ranging classesof stretchable electronics systems.

1. Introduction

Recent advances in mechanics and materials for stretchable/flexibleelectronics (Lacour et al., 2005; Khang et al., 2006; Lacour et al.,2006; Jiang et al., 2007; Jiang et al., 2008; Sekitani et al., 2009;Rogers et al., 2010; Huang et al., 2012; Yang and Lu, 2013; Duan et al.,2014) and optoelectronics (Kim et al., 2010; Lee et al., 2011a; Lipomiet al., 2011; Nelson et al., 2011) demonstrate that systems withhigh-performance semiconductor functionality can be realized in formsthat allow extreme mechanical deformations, e.g., stretching like arubber band, twisting like a rope, and bending like a sheet of paper.This class of technology creates many application opportunities thatcannot be addressed with established technologies, ranging from“epidermal” health/wellness monitors (Kim et al., 2011b; Kaltenbrunneret al., 2013; Schwartz et al., 2013), to soft surgical instruments(Cotton et al., 2009; Yu et al., 2009; Viventi et al., 2010; Graudejuset al., 2012; Kim et al., 2012b), to eyeball-like digital cameras (Ko etal., 2008; Song et al., 2013), to sensitive robotic skins (Someya etal., 2004; Wagner et al., 2004; Mannsfeld et al., 2010; Lu et al.,2012). Many of these stretchable systems exploit a strategy, sometimesknown as the island-bridge design (Kim et al., 2008; Ko et al., 2008;Kim et al., 2009; Kim et al., 2011 b; Lee et al., 2011b), in which theactive devices reside on non-deformable platforms (i.e. islands) withdeformable interconnects (i.e. bridges) in between. These bridgesprovide stretchability, while the islands undergo negligible deformation(usually <1% strain) to ensure mechanical integrity of the activedevices (Kim et al., 2008; Song et al., 2009). The stretchability of asystem with a certain filling ratio of islands can be written bystretchability of the system=(1−√{square root over (fillingratio)})*(stretchability of the interconnect).  (1)Various types of interconnect technologies have been developed,typically involving planar serpentines (Jones et al., 2004; Lacour etal., 2005; Li et al., 2005; Gonzalez et al., 2008; Kim et al., 2008; Hsuet al., 2009; Kim et al., 2011b; Kim et al., 2012c; Zhang et al., 2013c)or non-coplanar serpentines or straight bridges (Kim et al., 2008; Ko etal., 2008; Lee et al., 2011b). In many published examples, suchinterconnects offer total stretchability <50% (defined by onset ofcracks) and elastic stretchability <25% (defined by onset of plasticdeformation), in systems that do not significantly sacrifice the fillingratio. Many applications, particularly those in optoelectronics (Ko etal., 2008; Kim et al., 2010) and energy storage systems (Lipomi et al.,2011; Xu et al., 2013), also require high filling ratios. Here, advancedinterconnects are needed.

Recently, Xu et al. (2013) reported a design based on a type ofspace-filling curve that incorporates serpentine patterns in simplefractal-inspired layouts (shown in FIG. 54a ), to address theaforementioned challenges. The hierarchical structures with fractalinspired layouts have been shown to exist in many biological systems(Gao et al., 2005; Yao and Gao, 2006; Yao and Gao, 2007; Zhang et al.,2011; Li et al., 2012; Zhang et al., 2012; Li et al., 2013), which couldenhance or even control the surface adhesions, stiffness and materialstrengths. This technology, referred to initially as a ‘self-similar’design (Xu et al., 2013), enables stretchable lithium-ion batteries withtotal stretchability and elastic stretchability of ˜300% and ˜160%,respectively, and a filling ratio of ˜33%. The underlying mechanismsresponsible for this favorable mechanics were studied by bothexperiments and finite element analyses (FEA), as shown in FIG. 54b .The results reveal a mechanism of ordered unraveling. Specifically, withthe stretching proceeds from 0% to ˜150%, the 2^(nd) order structure(i.e., the large spring) first unravels via out-of-plane bending andtwisting through buckling, during which there is essentially nodeformation in the 1^(st) order structure (i.e., the small spring) (seetop 4 images, FIG. 54b ). The unraveling of the 1^(st) order structureonly starts as the 2^(nd) order structure is fully extended,corresponding to an applied strain of ˜150%. Additional, largestretchability (˜300%) is then achieved when the 1^(st) order structureis stretched to nearly its maximum extent (see bottom 3 images, FIG. 54b). Only the active materials are bonded to the soft substrate in thisbattery design such that the interconnects can deform freely. For somebiomedical applications (Kim et al., 2011 b; Kim et al., 2012c), theserpentine interconnects are either bonded to or encapsulated in thesoft substrate, and the resulting deformation mechanism may be quitedifferent from the free standing interconnects (Zhang et al., 2013b),but such aspects are beyond the scope of the present Example.

The filling ratio of active devices in the island-bridge design shown inFIG. 54a is 33%. The elastic stretchability (˜150%) is reduced to 22%and 4.3% for filling ratios of 90% and 98%, respectively. These levelsof elastic stretchability fall short of some biomedical applications,such as those in skin-mounted electronics (Kim et al., 2011b; Ying etal., 2012; Webb et al., 2013) and inflatable catheter technology (Kim etal., 2011a; Kim et al., 2012a), in which the strains (e.g., skin, heart,or elbow) may well exceed 20%. The most viable solution is to increasethe fractal order, from 2 in FIGS. 54a to 3 and 4 in FIG. 55 or evenhigher. For the fractal order of 4, however, the conventional FEAapproach becomes prohibitively time-consuming because of the largenumber of elements (>1 million) and the highly nonlinear postbucklinganalysis. Such a computational approach is impractical for rapid devicedesign and optimization.

The aim of the present Example is to develop an effective and robusthierarchical computational model (HCM), based on the mechanism ofordered unraveling illustrated in FIG. 54, for postbuckling analysis ofserpentine interconnects with fractal inspired layouts (referred to as“fractal interconnects” in the following). For an order-n fractalinterconnect under stretching, the lower order structures (≤n−1)initially do not unravel, and are only bent and twisted. As a result,these lower order structures can be modeled as straight beams witheffective tension, bending and torsion flexibilities, as illustrated inFIG. 56. Once the highest (n^(th)) order structure is fully stretched,unraveling of (n−1)^(th) order structure starts, but the (n−2)^(th) andlower order structures still do not unravel and can be modeled as beams.This process continues until the 1^(st) order structure unravels and thetotal stretchability is finally reached. Such an approach substantiallysaves computational effort because, at each order, only bending andtwisting of straight beams is involved. This simplification enablessimulations of high order (up to 4 as we demonstrated herein) fractalinterconnects, which would be quite difficult by using the conventionalFEA. This set of calculations not only illustrates the significanteffect of fractal order on stretchability, but also provides referencefor design using high-order fractal interconnects. The Example isoutlined as follows. Section 2 determines the equivalent flexibilitiesfor any order (n≥2) of fractal interconnect. Section 3 describes the HCMfor ordered unraveling of the postbuckling process. Section 4 appliesthe HCM to study the effect of fractal order on the elasticstretchability. Generalized fractal interconnects are studied in Section5, and the results are validated by experiments and conventional FEA.

2. Equivalent Flexibilities of Fractal Interconnects

2.1 Geometry

The 1^(st) order interconnect consists of straight wires and halfcircles that are connected in series, as shown in the black box of FIG.55, which has 4 unit cells in this example. The 2^(nd) orderinterconnect, shown in the blue box of FIG. 55, is created by reducingthe scale of the 1^(st) order interconnects, followed by 90° rotation,and then connecting them in a fashion that reproduces the layout of theoriginal geometry. The wide blue line in FIG. 55 represents the 2^(nd)order geometry that is similar to the 1^(st) order geometry (except forthe rounded part). By implementing the same procedure, we can generatethe 3^(rd) and 4^(th) order interconnects, as illustrated in the red andpurple boxes of FIG. 55, where the red and purple lines denote the3^(rd) and 4^(th) order geometries, respectively. It is clear that the2^(nd) to 4^(th) (and higher) order geometries all exhibit the samerectangular shape, and have the same number of unit cells, whichindicates that, strictly speaking, the fractal only starts at the 2^(nd)order.

Let η denote the height/spacing aspect ratio at each order such that theheight h^((i)) is related to the spacing l^((i)) of the i^(th) (i=1 . .. n) order (FIG. 55) by h^((i))=ηl^((i)). The height h^((i)) is alsorelated to the spacing i^((i−1)) of the neighboring order by the numberof unit cells m (FIG. 55) as h^((i))=2 ml^((i=1)) (i=2 . . . n). Thespacing and height at any order i are then scaled with the spacing ofthe highest order l^((n)) by

$\begin{matrix}{{l^{(i)} = {\left( \frac{\eta}{2\; m} \right)^{n - i}l^{(n)}}},{h^{(i)} = {{\eta\left( \frac{\eta}{2\; m} \right)}^{n - i}l^{(n)}}},{\left( {i = {1\mspace{14mu}\ldots\mspace{14mu} n}} \right).}} & (2)\end{matrix}$The result shows that a fractal interconnect is characterized by onebase length (l^((n))) and three non-dimensional parameters, namely thefractal order (n), the height/spacing ratio (η) and number (m) of unitcell.2.2. Equivalent Flexibilities

A fractal interconnect can be modeled as a beam if its width (w) andthickness (t) are much smaller than the length. FIG. 56a shows an n^(th)order fractal interconnect clamped at the left end, and subject toforces and bending moments at the right end. The axial force N, in-planeshear force Q_(y) and bending moment M_(z) at the right end induce thein-plane deformation represented by the displacements u_(x) and u_(y)and rotation θ_(z) at the end, while the out-of-plane shear force Q_(z)and bending moment M_(y), and torque M_(x) at the right end generate theout-of-plane displacement u_(z) and rotations θ_(x) and θ_(y) at theend. The normalized displacements, rotations, forces, bending momentsand torques are related by

$\begin{matrix}{{\begin{pmatrix}{u_{x}/l^{(n)}} \\{u_{y}/l^{(n)}} \\\theta_{z}\end{pmatrix} = {T_{{in} - {plane}}^{(n)}\mspace{11mu}\begin{pmatrix}{{N\left\lbrack l^{(n)} \right\rbrack}^{2}/({EI})_{{in} - {plane}}} \\{{Q_{y}\left\lbrack l^{(n)} \right\rbrack}^{2}/({EI})_{{in} - {plane}}} \\{M_{z}{l^{(n)}/({EI})_{{in} - {plane}}}}\end{pmatrix}}},} & \left( {3a} \right) \\{{\begin{pmatrix}{u_{z}/l^{(n)}} \\\theta_{y} \\\theta_{x}\end{pmatrix} = {T_{{out} - {of} - {plane}}^{(n)}\mspace{11mu}\begin{pmatrix}{{Q_{z}\left\lbrack l^{(n)} \right\rbrack}^{2}/({EI})_{{out} - {of} - {plane}}} \\{M_{y}{l^{(n)}/({EI})_{{out} - {of} - {plane}}}} \\{M_{x}{l^{(n)}/({EI})_{{out} - {of} - {plane}}}}\end{pmatrix}}},} & \left( {3b} \right)\end{matrix}$where (EI)_(in-plane)=Ew³t/12 and (EI)_(out-of-plane)=Ewt³/12 are thein-plane and out-of-plane bending stiffness, respectively, andT_(in-plane) ^((n)) and T_(out-of-plane) ^((n)) are the normalizedelastic flexibility matrices that can be obtained analytically (seeAppendix). For example, the in-plane flexibility matrix for the 1^(st)order fractal interconnect is

$\begin{matrix}{{{T_{{in} - {plane}}^{(1)}\left( {m,\eta} \right)} = \begin{Bmatrix}\begin{matrix}{\frac{m}{24}\left( {{4g^{3}} + {6\pi\; g^{2}} +} \right.} \\\left. {{24g} + {3\pi}} \right)\end{matrix} & \; & {Sym} \\{\frac{m}{4}\left( {g^{2} + {\pi\; g} + 2} \right)} & \begin{matrix}{{\frac{4m^{3}}{3}\left( {{2g} + \pi} \right)} +} \\{\frac{m}{24}\left( {{8g} + \pi} \right)}\end{matrix} & \; \\0 & {2\;{m^{2}\left( {g + \pi} \right)}} & {m\left( {{2g} + \pi} \right)}\end{Bmatrix}},} & \left( {4a} \right)\end{matrix}$where Sym denotes the symmetric matrix, and g=η−1. The out-of-planeflexibility matrix for the 1^(st) order fractal interconnect is

$\begin{matrix}{{{T_{{out} - {of} - {plane}}^{(1)}\left( {m,\eta} \right)} = \begin{Bmatrix}\begin{matrix}{{\frac{4m^{3}}{3}k} + {\frac{m}{48}\left\lbrack {{4k\left( {{3g^{2}} + 8} \right)} -} \right.}} \\\left. {{4\left( {1 + {3\; v}} \right)g^{3}} - {3\left( {7 + v} \right)\pi}} \right\rbrack\end{matrix} & \; & {Sym} \\{{- m^{2}}k} & {mk} & \; \\0 & 0 & {m\left\lbrack {k + {\left( {1 - v} \right)g}} \right\rbrack}\end{Bmatrix}},} & \left( {4b} \right)\end{matrix}$where v is the Poisson's ratio, and k=[4(1+v) g+(3+v)π]/4. For the2^(nd) order fractal interconnect, the in-plane flexibility matrix is

$\begin{matrix}{{{T_{{in} - {plane}}^{(2)}\left( {m,\eta} \right)} = \begin{Bmatrix}\begin{matrix}{{\frac{m}{4}{\eta^{2}\left( {4 - p} \right)}} +} \\{\frac{\eta^{3}}{4\; m^{2}}T_{{{in} - {plane}}\;,\; 22}^{(1)}}\end{matrix} & \; & {Sym} \\{\frac{m}{8}{\eta\left( {p + 2} \right)}} & \begin{matrix}{{\frac{4m^{3}}{3}p} + {\frac{m}{6}\left( {p - 2} \right)} +} \\{\frac{\eta^{3}}{4\; m^{2}}T_{{{in} - {plane}}\;,\; 11}^{(1)}}\end{matrix} & \; \\0 & {m^{2}p} & {mp}\end{Bmatrix}},} & \left( {5a} \right)\end{matrix}$where p=2η²+(π−2)η+2, T_(in-plane,11) ⁽¹⁾ and T_(in-plane,22) ⁽¹⁾ arethe 11 and 22 components in Eq. (4a), corresponding to the tensile andin-plane shear flexibilities, respectively.The out-of-plane flexibility matrix for the 2^(nd) order fractalinterconnect is

$\begin{matrix}{{{T_{{out} - {of} - {plane}}^{(2)}\left( {m,\eta} \right)} = \begin{Bmatrix}T_{{{out} - {of} - {plane}}\;,\; 11}^{(2)} & \; & {Sym} \\{- {m^{2}\left( {p - {\frac{1 - v}{4}{\pi\eta}}} \right)}} & {m\left( {p - {\frac{1 - v}{4}{\pi\eta}}} \right)} & \; \\0 & 0 & {m\left( {{\frac{1 - v}{4}p} + {\frac{1 - v}{4}{\pi\eta}}} \right)}\end{Bmatrix}},} & \left( {5b} \right)\end{matrix}$where T_(out-of-plane,11) ⁽²⁾ is given in Appendix. For the higher order(n≥3) fractal interconnects, T_(in-plane) ^((n)) and T_(out-of-plane)^((n)) are obtained in a recursive formula via the flexibility matricesfor the (n−1)^(th) and (n−2)^(th) order (see Appendix).

The flexibilities obtained from Eqs. (4) and (5) and the recursiveformula in the Appendix increase with the fractal order. For example,the tensile component T_(in-plane,11) ^((n)) (m=4, η=8/√{square rootover (11)}) increases from 15.4 for n=1, to 40.1 for n=2, and to 105 forn=3, i.e., by a factor of 2.6 for each increase of fractal order.Considering that the length l^((n)) also increases with n, thecorresponding tensile flexibility (without normalization) increases muchmore rapidly with the fractal order. For (EI)_(in-plane)=7.44×10⁻¹⁰ N·m²and l⁽¹⁾=110 μm as in the experiments (Xu et al., 2013), the tensileflexibility increases from 0.0276 m/N for n=1, to 2.62 m/N for n=2, andto 250 m/N for n=3, i.e., increasing by ˜100 times for each orderincrease. The fact that the flexibilities increase very rapidly with thefractal order will play a critical role in the development of the HCM inSection 3.

3. The Hierarchical Computational Model for Ordered Unraveling ofFractal Interconnects

As shown in FIG. 56a for 4 unit cells (m=4), an n^(th) order fractalinterconnect is composed of (n−1)^(th) order interconnects orientedalong vertical (y) direction, and (n−2)^(th) order interconnectsoriented along horizontal (x) direction. Before unraveling of any lowerorder interconnects, the (n−1)^(th) and (n−2)^(th) order interconnectsare modeled as the straight beams (in blue and orange colors,respectively, in FIG. 56b ) with the equivalent flexibilities T^((n−1))and T^((n−2)) obtained in Section 2. As shown in the following sections,such an approach gives accurate results, but the computation at eachorder is very simple since it involves only straight beams.

The 2^(nd) order fractal interconnect shown in FIG. 57 is used as anexample to illustrate the approach. The postbuckling process can beclassified into two stages.

(i) Stage I: Unraveling of the 2^(nd) order fractal interconnect asshown in FIG. 57a . The vertical, 1^(st) order fractal interconnects arerepresented by straight beams (blue color in FIG. 57a ) with theflexibilities given in Eq. (4). (The horizontal segments, denoted by thebrown color in FIG. 57a , are already straight beams.) FEA is used forthis equivalent structure of straight beams to determine the overallconfiguration under stretching. Stage I is complete when the equivalentstructure of straight beams is fully unraveled, i.e., the distancebetween the two ends reaches the total length of all straight beams. Theapplied strain, ε_(applied), defined by the percentage increase of thedistance between the two ends, reaches the critical value (ε_(cr(1))⁽²⁾) for a 2^(nd) order fractal interconnect at the end of stage I,

$\begin{matrix}{ɛ_{{cr}{(I)}}^{(2)} = {{\frac{m\left( {{2\; h^{(2)}} + {2\; l^{(2)}}} \right)}{2m\; l^{(2)}} - 1} = {\eta.}}} & (6)\end{matrix}$The initially vertical 1^(st) order fractal interconnects (blue color)become approximately horizontal (FIG. 4a ) at the end of stage I. Theirdeformations are essentially the same due to the periodicity of unitcells such that the analysis in stage II can focus on unraveling of asingle 1^(st) order fractal interconnect, as discussed in the following.

(ii) Stage II: Unraveling of each 1^(st) order fractal interconnect asshown in FIG. 57b . The stretching in stage II is mainly accommodated bythe (horizontally aligned) 1^(st) order fractal interconnects (bluecolor in FIG. 57a ) because their tensile flexibility is much largerthan that of straight beams (brown color). Thereby, the deformation ofthe straight beams is negligible, and only a single 1^(st) order fractalinterconnect (e.g., CD in FIG. 57b ) is analyzed by FEA (since all1^(st) order interconnects have essentially the same deformation), whichsubstantially reduces the computational cost. The additional stretchingin stage II, ε_(applied)−ε_(cr(1)) ⁽²⁾, corresponds to an additionaldisplacement 2 ml⁽²⁾[α_(applied)−ε_(cr(1)) ⁽²⁾] between the two ends ofthe 2^(nd) order fractal interconnect, which translates to thestretching displacement l⁽²⁾ [ε_(applied)−ε_(cr(1)) ⁽²⁾] in the FEA foreach 1^(st) order fractal interconnect in stage II. Stage II is completewhen each 1^(st) order interconnect is fully unraveled to reach itslength m[2h⁽¹⁾+(π−2)l⁽¹⁾].

The HCM introduced above is also applicable to higher orders (n≥3)fractal interconnects. For an order-n fractal interconnect, its(initially vertical) order-(n−1) and (horizontal) order-(n−2)interconnects are modeled as straight beams in stage I, followed byunraveling of order-(n−1) fractal interconnects in stage II. Allorder-(n−2) fractal interconnects, which result from both order-n andorder-(n−1) interconnects, start unraveling upon further stretchingafter stage II. This process repeats for all lower orders until the1^(st)-order fractal interconnects unravel.

4. Effect of Fractal Order on the Elastic Stretchability and Pattern ofDeformation

The HCM in Section 3 makes it possible to study the postbucklingbehavior of high order fractal interconnects with multiple unit cells,as shown in FIG. 55 for the 1^(st) to 4^(th) fractal interconnects withthe height/spacing aspect ratio η=8/√{square root over (11)} and numberof unit cell m=4. The copper interconnect has an elastic modulusE_(Cu)=119 GPa, Poisson's ratio v_(Cu)=0.34, and yield strain 0.3%(William et al., 1999) in an elastic-ideally plastic constitutive model(Hill, 1950)]. The results are validated by conventional FEA (withoutany approximations in the HCM) for the fractal order n≤3 because theanalyses of higher order (n≥4) interconnect would be extremely difficultand time-consuming by conventional FEA.

The elastic stretchability is the applied strain when the maximum strainin the interconnect reaches the yield strain (0.3%, William et al.(1999)) of the material. FIG. 58 shows the elastic stretchability versusthe order n of fractal interconnects for η=8/√{square root over (11)}and m=4. The thickness/width aspect ratio in the cross section ist/w=0.03, and the width to spacing ratio is w/l⁽¹⁾=0.4. The resultsagree very well with conventional FEA for n≤3. For each increase of n by1, the elastic stretchability increases by >3 times, suggesting that thehigh-order fractal design can substantially improve the elastic limit.For example, the elastic stretchability increases ˜200 times, from˜10.7% for the 1^(st) order, to 2140% for the 4^(th) order.

FIGS. 59-62 show the maximum principal strain in the fractalinterconnect and the evolution of deformation patterns for the fractalorder from 1 to 4, respectively. For the 1^(st) order fractalinterconnect (FIG. 59), the maximum principal strain increases rapidlywith the applied strain, and quickly reaches the yield strain 0.3%, atwhich the interconnect is still far from complete unraveling andtherefore leads to the elastic stretchability of only 10.7%. For the2^(nd) order fractal interconnect (FIG. 60a ), the maximum principalstrain initially increases slowly when the unraveling starts with the2^(nd) order structure, but then exhibits “strain hardening” near theend of unraveling (of the 2^(nd) order structure) (FIG. 60b ) for theapplied strain in range 150%<ε_(applied)<240%. The yield strain 0.3% isreached during the strain hardening, which gives 192% elasticstretchability. For the 3^(rd) order of fractal interconnect (FIG. 61a), there are two ranges of strain hardening, 150%<ε_(applied)<240% and500%<ε_(appl)<820%, corresponding to the (end of) unraveling of thehighest (3^(rd)) order and the next order (2^(nd)) structures,respectively (FIG. 61b ). The yield strain 0.3% is reached during thelatter strain hardening (corresponding to the unraveling of 2^(nd) orderstructures), which gives 747% elastic stretchability. As compared toFIG. 61a , the 4^(th) order fractal interconnect (FIG. 62a ) exhibits anadditional range of strain hardening (1500%<ε_(appl)<2300%), and itsthree ranges of strain hardening correspond to the (end of) unravelingof the 4^(th), 3^(rd) and 2^(nd) order structures, respectively (FIG.62b ). The elastic stretchability 2140% is reached during the laststrain hardening event (corresponding to the unraveling of 2^(nd) orderstructures). It is clear that the ordered unraveling of fractalinterconnects significantly retards the rate of increase of maximumprincipal strain, and therefore enables large elastic stretchability.

It should be pointed out that this level of interconnect stretchability(2140%) translates to 110% elastic stretchability of the system for ˜90%filling ratio of active devices based on Eq. (1), and 22% for 98%filling ratio, which are sufficient for biomedical applications.

5. Generalized Fractal Interconnects

The fractal interconnects discussed above can be generalized such thatat each order the interconnect may have its own height/spacing aspectratios η^((i)) and number of unit cells m^((i)) (i=1,2, . . . ,n). Thegeneralized fractal interconnects may provide simultaneously largeelastic stretchability and a relative low electrical resistance, asdemonstrated in Xu et al. (2013). For the n^(th) order generalizedfractal interconnect, the geometric relation (2) becomes

$\begin{matrix}{{l^{(i)} = {\left\lbrack {\prod\limits_{k = 1}^{n - i}\;\frac{\eta^{({n - k + 1})}}{2\; m^{({n - k})}}} \right\rbrack l^{(n)}}},{h^{(i)} = {{\eta^{(i)}\left\lbrack {\prod\limits_{k = 1}^{n - i}\;\frac{\eta^{({n - k + 1})}}{2\; m^{({n - k})}}} \right\rbrack}l^{(n)}}},{\left( {i = {{1\mspace{14mu}\ldots\mspace{14mu} n} - 1}} \right).}} & (7)\end{matrix}$

The HCM introduced in Section 3 can be extended straightforwardly tostudy the postbuckling of a generalized fractal interconnect. FIG. 63shows a generalized 2^(nd) order fractal interconnect in the experiments(Xu et al., 2013) of stretchable battery. The interconnect, as shown atthe top of FIG. 63 (e=0%), is composed of two polyimide (PI) layers(both 1.2 μm in thickness, E_(PI)=2.5 GPa and v_(PI)=0.34) that sandwichthe conducting copper layer (0.6 μm in thickness). The metal trace isconnected by two rigid, circular islands, which hardly deform during thestretching of the entire structure.

FIG. 63 compares the optical images from experiments to the resultsobtained by the HCM on the deformed configurations of the fractalinterconnect, for two different buckling modes, i.e., the symmetric andanti-symmetric modes. Good agreement between HCM and experiments areobserved over the entire range of stretching (0% to 300%) for bothmodes. The maximum strain in the metal layer obtained by the HCM agreeswell with that obtained by conventional FEA (FIG. 64), but the former ismuch faster and is applicable to higher fractal orders. The experiments(FIG. 63) and conventional FEA (FIG. 64) clearly validate the HCM.

6. Conclusions and Discussions

A hierarchical computational model for postbuckling analysis of fractalinterconnects based on the mechanism of ordered unraveling is developedin this Example. The approach substantially reduces the computationalefforts and costs compared to conventional FEA, but with accuratepredictions, as validated by both experiments and FEA. The designsprovide large enhancements (by ˜200 times) in the elasticstretchability, as the fractal order increases from 1 to 4. The HCM isalso applicable to many other types of fractal layouts formed withoutintersection points, such as Peano and Hilbert curves (Sagan, 1994). Itis useful for the development of stretchable electronics thatsimultaneously demand large areal coverage of active devices, such asstretchable photovoltaics (Yoon et al., 2008) and electronic eye-ballcameras (Ko et al., 2008). The concept of fractal interconnects can befurther combined with other strategies for stretchability, such asprestraining of soft substrate (Lee et al., 2011b; Zhang et al., 2013b),to further enhance the stretchability.

It should be pointed out that processes of ordered unraveling playcritical roles in the enhanced elastic stretchability of high-orderfractal interconnects, far beyond the simple increase of total length ofinterconnects with the fractal order. For example, the 1^(st) and 2^(nd)order fractal interconnects in FIG. 65 have the same total length andcross-section (width and thickness) and the same spacing between thedevice islands, yet the 2^(nd) order interconnect outperforms the 1^(st)order one in the elastic-stretchability by nearly a factor of 2 (528%versus 284%) (Xu et al., 2013).

Appendix. The Effective Flexibility of an n^(th) Order FractalInterconnect

For the 1^(st) order fractal interconnect clamped at the left end andsubject to forces (N, Q_(y), Q_(z)) and bending moments (M_(x), M_(y),M_(z)) at the right end, the strain energy of the entire interconnectcan be obtained from summation of the bending energy in all straight andcurved parts (Zhang et al., 2013a). The flexibility matricesT_(in-plane) ⁽¹⁾ and T_(out-of-plane) ⁽¹⁾ of the 1^(st) orderinterconnect can be obtained from the 2^(nd) order derivative of thestrain energy function with respect to the components of force (orbending moment), as given in Eqs. (4a) and (4b).

The 2^(nd) order fractal interconnect is composed of vertically aligned1^(st) order interconnects and horizontally aligned straight beams. Itsstrain energy is the sum of that in all 1^(st) order interconnects andstraight beams, whereas the strain energy of 1^(st) order interconnectcan be obtained using its flexibility matrices, T_(in-plane) ⁽¹⁾ andT_(out-of-plane) ⁽¹⁾. On the other hand, the strain energy of the 2^(nd)order fractal interconnect can be given in terms of the T_(in-plane) ⁽²⁾and T_(out-of-plane) ⁽²⁾. This energy equivalence leads to theanalytical expression of T_(in-plane) ⁽²⁾ and T_(out-of-plane) ⁽²⁾ inEqs. (5a) and (5b), where the component T_(out-of-plane,11) ⁽²⁾ is givenby

$\begin{matrix}{{T_{{{out} - {of} - {plane}}\;,\; 11}^{(2)} = {{\frac{m^{3}}{3}\left\lbrack {{4p} - {\left( {1 - v} \right){\pi\eta}}} \right\rbrack} + {\frac{m}{12}{\eta\left\lbrack {{k\left( {\eta^{2} + 2} \right)} + {\left( {5 + v} \right)\eta} - {2\left( {1 - v} \right)}} \right\rbrack}} + {\frac{\eta^{3}}{384\; m}\left\{ {{8\eta\; p} + {\left\lbrack {{\left( {5 + {3\; v}} \right)\pi} - 16} \right\rbrack\left( {p - {\pi\eta}} \right)} + {{8\left\lbrack {{8\left( {1 + v} \right)} - \pi} \right\rbrack}\eta} + {2\left( {7 + {5v}} \right)\pi} - {16\left( {3 + {4v}} \right)}} \right\}}}},} & \left( {A{.1}} \right)\end{matrix}$

An order-n interconnect is composed of vertically aligned order-(n−1)interconnects, and horizontally aligned order-(n−2) order interconnects.Based on the equivalence of strain energy of the order-n interconnectand that from summation of strain energy in all order-(n−1) andorder-(n−2) interconnects, T_(in-plane) ^((n)) and T_(out-of-plane)^((n)) are obtained in the following recursive formula via theflexibility matrices for the (n−1)^(th) and (n−2)^(th) order:

$\begin{matrix}{{T_{{in} - {plane}}^{(n)} = {{\frac{\eta}{2\; m}\begin{pmatrix}\frac{\eta}{2\; m} & 0 & 0 \\0 & \frac{\eta}{2\; m} & 0 \\0 & 0 & 1\end{pmatrix}{\sum\limits_{k = 1}^{m}\;{\begin{Bmatrix}\begin{matrix}{{\overset{\_}{D}}_{I}\left\lbrack {{T_{{in} - {plane}}^{({n - 1})}\mspace{11mu} K_{1}(m)} +} \right.} \\{{\left. {{K_{1}^{T}(m)}T_{{in} - {plane}}^{({n - 1})}} \right\rbrack{\overset{\_}{D}}_{I}^{T}} +}\end{matrix} \\{{{\overset{\_}{D}}_{II}T_{{in} - {plane}}^{({n - 1})}\mspace{11mu}{\overset{\_}{D}}_{II}^{T}} +} \\\begin{matrix}{{\overset{\_}{D}}_{III}\left\lbrack {{T_{{in} - {plane}}^{({n - 1})}\mspace{11mu} K_{1}(m)} +} \right.} \\{\left. {{K_{1}^{T}(m)}T_{{in} - {plane}}^{({n - 1})}} \right\rbrack{\overset{\_}{D}}_{III}^{T}}\end{matrix}\end{Bmatrix}\begin{pmatrix}\frac{\eta}{2\; m} & 0 & 0 \\0 & \frac{\eta}{2\; m} & 0 \\0 & 0 & 1\end{pmatrix}}}} + {\frac{\eta^{2}}{4\; m^{2}}{\sum\limits_{k = 1}^{m}\left\lbrack {{{\overset{\_}{D}}_{IV}^{\;*}T_{{in} - {plane}}^{({n - 2})}\mspace{14mu}{\overset{\_}{D}}_{IV}^{\;*}} + {{\overset{\_}{D}}_{V}^{\;*}T_{{in} - {plane}}^{({n - 2})}\mspace{11mu}{\overset{\_}{D}}_{V}^{\;{*T}}}} \right\rbrack}}}}\;\mspace{79mu}{{{{for}\mspace{14mu} n} \geq 3},}} & \left( {A{.2}a} \right) \\{{T_{{out} - {of} - {plane}}^{(n)} = {{\frac{\eta}{2\; m}\begin{pmatrix}\frac{\eta}{2\; m} & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{pmatrix}{\sum\limits_{k = 1}^{m}\;{\begin{Bmatrix}\begin{matrix}{{\overset{\_}{R}}_{I}\left\lbrack {{T_{{out} - {of} - {plane}}^{({n - 1})}\mspace{11mu} K_{2}(m)} +} \right.} \\{{\left. {{K_{2}^{T}(m)}T_{{out} - {of} - {plane}}^{({n - 1})}} \right\rbrack{\overset{\_}{R}}_{I}^{T}} +}\end{matrix} \\{{{\overset{\_}{R}}_{II}T_{{out} - {of} - {plane}}^{({n - 1})}\mspace{11mu}{\overset{\_}{R}}_{II}^{T}} +} \\\begin{matrix}{{\overset{\_}{R}}_{III}\left\lbrack {{T_{{out} - {of} - {plane}}^{({n - 1})}\mspace{11mu} K_{2}(m)} +} \right.} \\{\left. {{K_{2}^{T}(m)}T_{{out} - {of} - {plane}}^{({n - 1})}} \right\rbrack{\overset{\_}{R}}_{III}^{T}}\end{matrix}\end{Bmatrix}\begin{pmatrix}\frac{\eta}{2\; m} & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{pmatrix}}}} + {\frac{\eta^{2}}{4\; m^{2}}{\sum\limits_{k = 1}^{m}\left\lbrack {{{\overset{\_}{R}}_{IV}^{\;*}T_{{out} - {of} - {plane}}^{({n - 2})}\mspace{11mu}{\overset{\_}{R}}_{IV}^{\;*}} + {{\overset{\_}{R}}_{V}^{\;*}T_{{out} - {of} - {plane}}^{({n - 2})}\mspace{11mu}{\overset{\_}{R}}_{V}^{\;{*T}}}} \right\rbrack}}}}\;\mspace{79mu}{{{{for}\mspace{14mu} n} \geq 3},}} & \left( {A{.2}b} \right) \\{where} & \; \\\; & \; \\{{{\overset{\_}{D}}_{I} = \begin{bmatrix}0 & 1 & {- m} \\1 & 0 & {{- 4}\left( {m - k + 1} \right)m\;\eta^{- 1}} \\0 & 0 & {- 1}\end{bmatrix}},} & \; \\{{{\overset{\_}{D}}_{II} = \begin{bmatrix}0 & 1 & {- m} \\{- 1} & 0 & {\left( {{4m} - {4k} + 2} \right)m\;\eta^{- 1}} \\0 & 0 & 1\end{bmatrix}},} & \; \\{{{\overset{\_}{D}}_{III} = \begin{bmatrix}0 & 1 & 0 \\1 & 0 & {{- 4}\left( {m - k} \right)m\;\eta^{- 1}} \\0 & 0 & {- 1}\end{bmatrix}},} & \; \\{{{\overset{\_}{D}}_{IV}^{*} = \begin{bmatrix}{\eta^{2}/\left( {4\; m^{2}} \right)} & 0 & {\eta/2} \\0 & {\eta^{2}/\left( {4\; m^{2}} \right)} & {{2m} - {2k} + 1} \\0 & 0 & 1\end{bmatrix}},} & \; \\{{{\overset{\_}{D}}_{V}^{*} = \begin{bmatrix}{\eta^{2}/\left( {4\; m^{2}} \right)} & 0 & {\eta/2} \\0 & {{- \eta^{2}}/\left( {4\; m^{2}} \right)} & {{{- 2}m} + {2k}} \\0 & 0 & {- 1}\end{bmatrix}},} & \; \\{{{K_{1}(m)} = {\frac{1}{4}\begin{pmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & {- m} & 1\end{pmatrix}}},} & \; \\{{{\overset{\_}{R}}_{I} = \begin{bmatrix}1 & m & {{- 4}\left( {m - k + 1} \right)m\;\eta^{- 1}} \\0 & 0 & 1 \\0 & {- 1} & 0\end{bmatrix}},} & \; \\{{{\overset{\_}{R}}_{II} = \begin{bmatrix}1 & m & {\left( {{4m} - {4k} + 2} \right)m\;\eta^{- 1}} \\0 & 0 & {- 1} \\0 & 1 & 0\end{bmatrix}},} & \; \\{{{\overset{\_}{R}}_{III} = \begin{bmatrix}1 & 0 & {{- 4}\left( {m - k} \right)m\;\eta^{- 1}} \\0 & 0 & 1 \\0 & {- 1} & 0\end{bmatrix}},} & \; \\{{{\overset{\_}{R}}_{IV}^{*} = \begin{bmatrix}{\eta^{2}/\left( {4\; m^{2}} \right)} & {- \left( {{2m} - {2k} + 1} \right)} & {{- \eta^{3}}/\left( {8\; m^{2}} \right)} \\0 & {\eta^{2}/\left( {4\; m^{2}} \right)} & 0 \\0 & 0 & {\eta^{2}/\left( {4\; m^{2}} \right)}\end{bmatrix}},} & \; \\{{{\overset{\_}{R}}_{V}^{*} = \begin{bmatrix}{\eta^{2}/\left( {4\; m^{2}} \right)} & {- \left( {{2m} - {2k}} \right)} & {{- \eta^{3}}/\left( {8\; m^{2}} \right)} \\0 & {\eta^{2}/\left( {4\; m^{2}} \right)} & 0 \\0 & 0 & {{- \eta^{2}}/\left( {4\; m^{2}} \right)}\end{bmatrix}},} & \; \\{{{and}\mspace{20mu}{K_{2}(m)}} = {\frac{1}{4}{\begin{pmatrix}1 & 0 & 0 \\m & 1 & 0 \\0 & 0 & 1\end{pmatrix}.}}} & \;\end{matrix}$

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STATEMENTS REGARDING INCORPORATION BY REFERENCE AND VARIATIONS

All references throughout this application, for example patent documentsincluding issued or granted patents or equivalents; patent applicationpublications; and non-patent literature documents or other sourcematerial; are hereby incorporated by reference herein in theirentireties, as though individually incorporated by reference, to theextent each reference is at least partially not inconsistent with thedisclosure in this application (for example, a reference that ispartially inconsistent is incorporated by reference except for thepartially inconsistent portion of the reference).

The terms and expressions which have been employed herein are used asterms of description and not of limitation, and there is no intention inthe use of such terms and expressions of excluding any equivalents ofthe features shown and described or portions thereof, but it isrecognized that various modifications are possible within the scope ofthe invention claimed. Thus, it should be understood that although thepresent invention has been specifically disclosed by preferredembodiments, exemplary embodiments and optional features, modificationand variation of the concepts herein disclosed may be resorted to bythose skilled in the art, and that such modifications and variations areconsidered to be within the scope of this invention as defined by theappended claims. The specific embodiments provided herein are examplesof useful embodiments of the present invention and it will be apparentto one skilled in the art that the present invention may be carried outusing a large number of variations of the devices, device components,methods steps set forth in the present description. As will be obviousto one of skill in the art, methods and devices useful for the presentmethods can include a large number of optional composition andprocessing elements and steps.

When a group of substituents is disclosed herein, it is understood thatall individual members of that group and all subgroups, including anyisomers, enantiomers, and diastereomers of the group members, aredisclosed separately. When a Markush group or other grouping is usedherein, all individual members of the group and all combinations andsubcombinations possible of the group are intended to be individuallyincluded in the disclosure. When a compound is described herein suchthat a particular isomer, enantiomer or diastereomer of the compound isnot specified, for example, in a formula or in a chemical name, thatdescription is intended to include each isomers and enantiomer of thecompound described individual or in any combination. Additionally,unless otherwise specified, all isotopic variants of compounds disclosedherein are intended to be encompassed by the disclosure. For example, itwill be understood that any one or more hydrogens in a moleculedisclosed can be replaced with deuterium or tritium. Isotopic variantsof a molecule are generally useful as standards in assays for themolecule and in chemical and biological research related to the moleculeor its use. Methods for making such isotopic variants are known in theart. Specific names of compounds are intended to be exemplary, as it isknown that one of ordinary skill in the art can name the same compoundsdifferently.

Many of the molecules disclosed herein contain one or more ionizablegroups [groups from which a proton can be removed (e.g., —COOH) or added(e.g., amines) or which can be quaternized (e.g., amines)]. All possibleionic forms of such molecules and salts thereof are intended to beincluded individually in the disclosure herein. With regard to salts ofthe compounds herein, one of ordinary skill in the art can select fromamong a wide variety of available counterions those that are appropriatefor preparation of salts of this invention for a given application. Inspecific applications, the selection of a given anion or cation forpreparation of a salt may result in increased or decreased solubility ofthat salt.

Every formulation or combination of components described or exemplifiedherein can be used to practice the invention, unless otherwise stated.

It must be noted that as used herein and in the appended claims, thesingular forms “a”, “an”, and “the” include plural reference unless thecontext clearly dictates otherwise. Thus, for example, reference to “acell” includes a plurality of such cells and equivalents thereof knownto those skilled in the art, and so forth. As well, the terms “a” (or“an”), “one or more” and “at least one” can be used interchangeablyherein. It is also to be noted that the terms “comprising”, “including”,and “having” can be used interchangeably. The expression “of any ofclaims XX-YY” (wherein XX and YY refer to claim numbers) is intended toprovide a multiple dependent claim in the alternative form, and in someembodiments is interchangeable with the expression “as in any one ofclaims XX-YY.”

Whenever a range is given in the specification, for example, atemperature range, a time range, or a composition or concentrationrange, all intermediate ranges and subranges, as well as all individualvalues included in the ranges given are intended to be included in thedisclosure. As used herein, ranges specifically include the valuesprovided as endpoint values of the range. For example, a range of 1 to100 specifically includes the end point values of 1 and 100. It will beunderstood that any subranges or individual values in a range orsubrange that are included in the description herein can be excludedfrom the claims herein.

All patents and publications mentioned in the specification areindicative of the levels of skill of those skilled in the art to whichthe invention pertains. References cited herein are incorporated byreference herein in their entirety to indicate the state of the art asof their publication or filing date and it is intended that thisinformation can be employed herein, if needed, to exclude specificembodiments that are in the prior art. For example, when composition ofmatter are claimed, it should be understood that compounds known andavailable in the art prior to Applicant's invention, including compoundsfor which an enabling disclosure is provided in the references citedherein, are not intended to be included in the composition of matterclaims herein.

As used herein, “comprising” is synonymous with “including,”“containing,” or “characterized by,” and is inclusive or open-ended anddoes not exclude additional, unrecited elements or method steps. As usedherein, “consisting of” excludes any element, step, or ingredient notspecified in the claim element. As used herein, “consisting essentiallyof” does not exclude materials or steps that do not materially affectthe basic and novel characteristics of the claim. In each instanceherein any of the terms “comprising”, “consisting essentially of” and“consisting of” may be replaced with either of the other two terms. Theinvention illustratively described herein suitably may be practiced inthe absence of any element or elements, limitation or limitations whichis not specifically disclosed herein.

One of ordinary skill in the art will appreciate that startingmaterials, biological materials, reagents, synthetic methods,purification methods, analytical methods, assay methods, and biologicalmethods other than those specifically exemplified can be employed in thepractice of the invention without resort to undue experimentation. Allart-known functional equivalents, of any such materials and methods areintended to be included in this invention. The terms and expressionswhich have been employed are used as terms of description and not oflimitation, and there is no intention that in the use of such terms andexpressions of excluding any equivalents of the features shown anddescribed or portions thereof, but it is recognized that variousmodifications are possible within the scope of the invention claimed.Thus, it should be understood that although the present invention hasbeen specifically disclosed by preferred embodiments and optionalfeatures, modification and variation of the concepts herein disclosedmay be resorted to by those skilled in the art, and that suchmodifications and variations are considered to be within the scope ofthis invention as defined by the appended claims.

We claim:
 1. A electronic circuit comprising: an elastic substrate; anda stretchable metallic or semiconducting device component supported bysaid elastic substrate; said stretchable metallic or semiconductingdevice component comprising a plurality of electrically conductiveelements each having a primary unit cell shape, said electricallyconductive elements connected in a sequence having a secondary shapeproviding an overall two-dimensional spatial geometry characterized by aplurality of spatial frequencies, wherein said two-dimensional spatialgeometry is a self-similar two-dimensional geometry; wherein saidtwo-dimensional spatial geometry of said metallic or semiconductingdevice component allows for accommodation of elastic strain along one ormore in-plane or out of plane dimensions, thereby providingstretchability of said electronic circuit, and wherein said sequence ofelectrically conductive elements is further characterized by a tertiaryshape comprising a repeating series of said electrically conductiveelements comprising said secondary shape.
 2. The electronic circuit ofclaim 1, wherein said two-dimensional spatial geometry allows saidmetallic or semiconducting device component to undergo elasticdeformation.
 3. The electronic circuit of claim 1, wherein saidtwo-dimensional spatial geometry is characterized by a first spatialfrequency having a first length scale corresponding to said primary unitcell shape and a second spatial frequency having a second length scalecorresponding to said secondary shape, and wherein said first lengthscale of said first spatial frequency is selected from the range of 100nm to 1 mm and wherein said second length scale of said second spatialfrequency is selected from the range of 1 micron to 10 mm.
 4. Theelectronic circuit of claim 1, wherein said two-dimensional spatialgeometry has a spring-within-a-spring geometry, wherein saidspring-within-in-spring geometry comprises a series of primary springstructures each independently having said primary unit cell shapeconnected to form one or more secondary spring structures having saidsecondary shape.
 5. The electronic circuit of claim 1, wherein a saidtwo-dimensional spatial geometry of said stretchable metallic orsemiconducting device component provides a fill factor between first andsecond device components or provided over an active area of saidelectronic circuit selected from the range of 20% to 90%.
 6. Theelectronic circuit claim 1, wherein said electrically conductiveelements of said metallic or semiconducting device component comprise acontinuous structure or a single unitary structure.
 7. The electroniccircuit of claim 1, wherein each of said electrically conductiveelements independently has a thickness less than or equal to 1 micron.8. The electronic circuit of claim 1, wherein said electricallyconductive elements independently comprise a metal, an alloy, a singlecrystalline inorganic semiconductor or an amorphous inorganicsemiconductor.
 9. The electronic circuit of claim 1, wherein saidprimary unit cell shape of said electrically conductive elementscomprises a spring, a fold, a loop, a mesh or any combinations of these.10. The electronic circuit of claim 1, wherein said stretchable metallicor semiconducting device component comprises an electrode or anelectrode array.
 11. The electronic circuit of claim 10, wherein saidelectrode or said electrode array is a component of sensor, actuator, ora radio frequency device.
 12. The electronic circuit of claim 1,comprising one or more rigid island structures, wherein said stretchablemetallic or semiconducting device component comprises one or moreelectrical interconnects, wherein at least a portion of said one or moreelectrical interconnects is in electrical contact with said rigid islandstructures.
 13. The electronic circuit of claim 12, wherein said rigidisland structures comprise inorganic semiconductor devices or devicecomponents and wherein said inorganic semiconductor devices or devicecomponents comprise a transistor, a diode, an amplifier, a multiplexer,a light emitting diode, a laser, a photodiode, an integrated circuit, asensor, a temperature sensor, a thermistor, a heater, a resistiveheater, an actuator or any combination of these.
 14. The electroniccircuit of claim 1, wherein said elastic substrate has an averagethickness less than or equal to 1000 μm.
 15. The electronic circuit ofclaim 1, wherein said elastic substrate has a Young's modulus selectedfrom the range of 0.5 KPa to 100 Gpa and a net net bending stiffnessselected from the range of 0.1×10⁴ GPa μm⁴ to 1×10⁹ GPa μm⁴.
 16. Theelectronic circuit of claim 1, wherein said elastic substrate comprisesa material selected from the group consisting of: a polymer, aninorganic polymer, an organic polymer, a plastic, an elastomer, abiopolymer, a thermoset, a rubber silk and any combination of these. 17.The electronic circuit of claim 1, comprising an energy storage device,a photonic device, an optical sensor, a strain sensor, an electricalsensor, a temperature sensor, a chemical sensor, an actuator, acommunication device, a micro- or nano-fluidic device, an integratedcircuit or any component thereof.
 18. The electronic circuit of claim 1,comprising a tissue mounted electronic device, a radio frequency antennaor a sensor compatible with magnetic resonance imaging.
 19. An electrodearray comprising: a plurality of stretchable metallic or semiconductingdevice components supported by said elastic substrate, wherein each ofsaid stretchable metallic or semiconducting device componentsindependently comprises a plurality of electrically conductive elementseach having a primary unit cell shape, wherein said electricallyconductive elements of each stretchable metallic or semiconductingdevice component are independently connected in a sequence having asecondary shape providing an overall two-dimensional spatial geometrycharacterized by a plurality of spatial frequencies, wherein saidtwo-dimensional spatial geometry is a self-similar two-dimensionalgeometry; wherein said sequence of electrically conductive elements isfurther characterized by a tertiary shape comprising a repeating seriesof said electrically conductive elements comprising said secondaryshape.
 20. A stretchable electronic device comprising: a plurality ofrigid island structures supported by an elastic substrate; wherein eachof said rigid island structures independently comprises an inorganicsemiconductor device or device component; a plurality of stretchablemetallic or semiconducting device components electricallyinterconnecting at least a portion of said rigid island structures,wherein each of said stretchable metallic or semiconducting devicecomponents independently comprises a plurality of electricallyconductive elements each having a primary unit cell shape, wherein saidelectrically conductive elements of each stretchable metallic orsemiconducting device component are independently connected in asequence having a secondary shape providing an overall two-dimensionalspatial geometry characterized by a plurality of spatial frequencies,wherein said two-dimensional spatial geometry is a self-similartwo-dimensional geometry; wherein said two-dimensional spatialgeometries of said metallic or semiconducting device components allowsfor accommodation of elastic strain along one or more in-plane or out ofplane dimensions, thereby providing stretchability of said stretchableelectronic device; and wherein said sequence of electrically conductiveelements is further characterized by a tertiary shape comprising arepeating series of said electrically conductive elements comprisingsaid secondary shape.